Holomaps

Holomaps, or holomorphic maps, are functions between complex manifolds that preserve their complex structure. Let X and Y be complex manifolds. A map f: X -> Y is holomorphic if, for every point x in X, there exist local charts around x and around f(x) such that the coordinate representation of f is holomorphic between open subsets of complex Euclidean spaces. Equivalently, when expressed in local coordinates, the component functions of f are holomorphic.

Key properties of holomorphic maps include stability under composition and the fact that the identity map is holomorphic. Holomorphic maps are continuous and preserve the complex-analytic structure, in particular pulling back holomorphic functions to holomorphic functions. Many familiar objects are holomorphic maps: ordinary holomorphic functions on open subsets of complex Euclidean space, as well as polynomial and rational maps between complex varieties.

In one complex variable, foundational results such as the open mapping theorem and the maximum modulus principle have as consequences that nonconstant holomorphic maps are open and exhibit rigidity. In several complex variables, analogous themes persist, though the theory is richer and more nuanced due to higher-dimensional phenomena.

Special topics in holomaps include proper and finite holomorphic maps. A map is proper if preimages of compact sets are compact; such maps often behave like finite coverings and can be described using ramification data. In the setting of Riemann surfaces (one complex dimension), holomorphic maps are classified by degree and branch points, reflecting their geometric multiplicity.

Overall, holomaps form the morphisms in the category of complex manifolds and are central to complex geometry and the study of several complex variables.