Work locally, suppose X and Y are open subsets of C^n where C is the complex number field. Suppose f: X—>Y is a map given by n polynomials. If f is quasi-finite (i.e. each fiber is a finite set) and surjective, then is f an open map?
Another question is about finite morphism in the analytic category. Let X and Y be complex manifolds and f:X—>Y an analytic (i.e. holomorphic) map. I guess there is a notion of f being finite, and a definition is that the induced maps between the local rings of germs of analytic functions are finite ring homomorphisms. Are there equivalent and more transparent characterizations of f being finite?
Best Answer
Open Mapping Theorem [Grauert-Remmert: Coherent analytic sheaves, p.107] Let $X,Y$ be pure $d$-dimensional complex spaces and assume that $Y$ is locally irreducible. Then any holomorphic map $f:X\to Y$ with discrete fibers is open.