# The Moduli Space of smooth Ample Hypersurfaces in Abelian Varieties

## TL;DR

This paper characterizes smooth ample hypersurfaces in Abelian varieties and describes a specific irreducible component of their moduli space, including hypersurfaces of a fixed polarization type and certain coverings, also relating to Kähler structures.

## Contribution

It provides a detailed description of the moduli space component for smooth ample hypersurfaces in Abelian varieties, including their polarization types and coverings.

## Key findings

- Identifies a connected component of the moduli space for these hypersurfaces.
- Describes the structure of hypersurfaces with a fixed polarization type.
- Links the moduli space component to Kähler complex structures.

## Abstract

We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated univariate coverings of normal type (of the same polarization type). The above manifolds yield also a connected component of the open set of Teichm\"uller space consisting of K\"ahler complex structures.