A Kähler manifold is a special type of complex manifold that has a compatible symplectic structure and a Riemannian metric. This means it has both complex and geometric properties that work well together. The manifold is equipped with a Kähler metric, which allows for the measurement of distances and angles while preserving the complex structure. Kähler manifolds are important in various fields of mathematics and theoretical physics, particularly in string theory and algebraic geometry. They provide a rich framework for studying complex geometry and have applications in understanding the geometry of Calabi-Yau manifolds, which are crucial for compactifying extra dimensions in string theory.