What is a schlicht domain over $\mathbb{C}^n$? How is it different from a domain in $\mathbb{C}^n$? Examples?
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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.
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1$\begingroup$ More on the "schlicht property": mathoverflow.net/questions/62218/… $\endgroup$ Commented Nov 22, 2012 at 22:33
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1$\begingroup$ So, every domain in $\mathbb C^n$ is schlicht. $\endgroup$ Commented Nov 22, 2012 at 22:50