An Abelian surface is a type of complex algebraic variety that is a higher-dimensional generalization of an elliptic curve. Specifically, it is a two-dimensional Abelian variety, which means it has a group structure that is compatible with its geometric properties. Abelian surfaces can be defined over various fields, including the complex numbers, and they play a significant role in algebraic geometry and number theory. These surfaces can be represented as a quotient of a complex torus by a lattice, and they exhibit rich structures, including a duality between points and divisors. Abelian surfaces are also connected to various mathematical concepts, such as moduli spaces and automorphisms, making them important in the study of algebraic geometry and arithmetic geometry.