What is a *schlicht* domain over $\mathbb{C}^n$? How is it different from a domain in $\mathbb{C}^n$? Examples?



Schlicht domain
*Schlicht domain* over ${\mathbb C}^n$ is the same as a domain in ${\mathbb C}^n$. The point is that one also defines *domains over ${\mathbb C}^n$* as connected complex manifolds $M^n$ equipped with a locally biholomorphic map $f:M^n\to {\mathbb C}^n$. The *schlicht* property just means that $f$ is 1-1. 


See <http://www.encyclopediaofmath.org/index.php/Riemannian_domain> for general definition and references for Riemann domains. People also consider *branched* Riemann domains where "locally biholomorphic" is replaced with "holomorphic with discrete fibers." These are generalizations of Riemann surfaces of multivalued holomorphic functions of one variable. 


