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 equipped with additional structures, such as a Kähler metric, which provides a way to measure distances and angles. They also play a crucial role in theoretical physics, particularly in understanding the geometry of Calabi-Yau manifolds, which are important in string theory and compactification.