A Kähler manifold is a special type of complex manifold that has a compatible Riemannian metric and symplectic structure. This means it has both a complex structure, allowing for complex coordinates, and a way to measure distances and angles, which is provided by the Riemannian metric. The symplectic structure is a non-degenerate, closed 2-form that helps in understanding the geometry and topology of the manifold. Kähler manifolds are important in various areas of mathematics and theoretical physics, particularly in string theory and algebraic geometry. They exhibit rich geometric properties, such as being both Kähler and Ricci-flat, which leads to significant implications in the study of Calabi-Yau manifolds and their applications in compactifying extra dimensions in physics.