The Infinitesimal Torelli problem for hypersurfaces in simple abelian varieties

Given a family of compact 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 manifolds. I will prove it for smooth hypersurfaces in simple abelian varieties with sufficiently high self intersection giving an effective bound on a result by Green in this particular case. If time permits I will also outline what happens in the non-simple case.