Holomorphic Mappings: Survey of Some Results and Discussion of Open Problems

1. Introduction. Our purpose is to survey some aspects of the global theory of holomorphic mappings, isolating along the way a few outstanding questions. The general problem is this : Let M and N be complex manifolds and ƒ \M-> N a holomorphic mapping. Study the position of the image f(M) in N. In particular, if S c N is a complex analytic subvariety, then setting S f = f~ l (S\ study the S f c M as S varies among the subvarieties of N. The most important special case of this problem is when M = C m and N = P n , the complex projective space. Then ƒ may be given by n entire meromorphic functions