The moduli space of smooth ample hypersurfaces in Abelian varieties

We give a characterization of smooth ample Hypersurfaces in Abelian Varieties and also describe the connected component of the moduli space containing such hypersurfaces of a given polarization type: we show that this component is irreducible and that it consists of these Hypersurfaces, plus the smooth iterated univariate coverings of normal type (of the same polarization type). The above manifolds yield also a connected component of the open set of Teichmfiller space consisting of Kahler complex structures. (C) 2020 Elsevier Inc. All rights reserved.