The Infinitesimal Torelli Theorem for Irregular Varieties

Abstract

In this thesis we prove the infinitesimal Torelli theorem for certain classes of irregular varieties. Given a compact Kähler manifold, the infinitesimal Torelli problem asks whether the differential of the period map of a Kuranishi family is injective. Unlike the classical Torelli theorem for curves, there is a negative answer for example for hyperelliptic curves of genus greater than 2. Nevertheless, the infinitesimal Torelli theorem holds for many other classes of manifolds. Following Green’s proof for sufficiently ample hypersurfaces in arbitrary varieties, we prove it for smooth ample hypersurfaces and more generally complete intersections in general abelian varieties by reducing it to showing the surjectivity of certain multiplication maps of vector bundles on the ambient abelian variety. Then we derive numerical conditions for such multiplication maps to be surjective giving an effective bound on Green’s result in this particular case. We also investigate the more general case of irregular varieties with globally generated cotangent bundle which do not embed into their Albanese varieties.