Complex manifold

A complex manifold is a manifold whose coordinate charts are modeled on open subsets of C^n rather than R^n, with transition functions that are holomorphic rather than merely smooth. This seemingly small change — replacing real differentiability with complex analyticity — imposes severe constraints on the topology and geometry of the manifold. Unlike real manifolds, which are locally all alike, complex manifolds carry intrinsic invariants (Hodge numbers, Chern classes) that distinguish them at the most local level. A Kähler manifold is a complex manifold that admits a compatible Riemannian metric and symplectic structure, but most complex manifolds are not Kähler. The study of which complex manifolds satisfy the Kähler condition, and why, remains one of the central problems of complex geometry.