Holomorphic map between complex manifolds ($F:M⟶N$)

A map between complex manifolds is holomorphic when its local coordinate expression has holomorphic scalar components. For a smooth map, this is equivalent to its differential being complex-linear at every point. These maps compose by the chain rule. For one-dimensional manifolds this reduces to a holomorphic map between Riemann surfaces; the general definition also applies to multiplication and inversion on a complex Lie group.