The infinitesimal Torelli problem for ample hypersurfaces in abelian varieties

Given a family of Kähler manifolds, the infinitesimal Torelli problem asks whether the differential of the period map is injective. Unlike the classical Torelli theorem, it fails even for certain curves (specifically hyperelliptic curves of genus >2). Nevertheless it holds for many other classes of objects. Griffiths proved it for hypersurfaces in projetive space and Green generalized this to sufficiently ample hypersurfaces in arbitrary varieties. However, he does not give an effective bound on the required ampleness. In this talk I will give an overview of the problem and then consider the special case of hypersurfaces in abelian varieties. The infinitesimal Torelli can be reduced to proving the surjectivity of a certain multiplication map of global sections of line bundles. I will give explicit criteria in some cases for the infinitesimal Torelli to hold in terms of the type of the polarization of the ambient abelian variety defined by the hypersurface.