A complex manifold is a mathematical structure that generalizes the concept of complex numbers to higher dimensions. It consists of a set of points that locally resemble complex Euclidean space, allowing for the study of complex functions and geometric properties. Complex manifolds are essential in various fields, including algebraic geometry and string theory. These manifolds can be thought of as spaces where complex coordinates can be used, making them richer than real manifolds. They are equipped with a specific type of structure called a holomorphic structure, which enables the definition of complex differentiable functions. This property is crucial for understanding the behavior of complex systems.