An abelian variety is a type of algebraic variety that is also a group, meaning it has a structure that allows for addition of its points. These varieties are defined over fields and are characterized by their complex structure, which allows them to be studied using both algebraic and geometric methods. They play a crucial role in number theory and algebraic geometry. Abelian varieties can be thought of as higher-dimensional generalizations of elliptic curves, which are the simplest examples of abelian varieties. They are used in various areas of mathematics, including the study of modular forms and cryptography. Their rich structure allows mathematicians to explore deep connections between different areas of mathematics.