Complex differential geometry is a branch of mathematics that studies geometric structures on complex manifolds, which are spaces that locally resemble complex Euclidean space. It combines techniques from both differential geometry and complex analysis, focusing on properties that arise from the interplay between these two fields. One key concept in complex differential geometry is the notion of a Kähler manifold, which is a special type of complex manifold equipped with a symplectic form that is compatible with the complex structure. This area has applications in various fields, including theoretical physics and string theory, where the geometry of complex spaces plays a crucial role.