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Edited biholomorphic function to follow the same format as diffeomorphism. In particular, this means that I qualified biholomorphic function to refer only to maps between complex manifolds. Is there a more general definition of holomorphic functions between complex analytic spaces?
Isn’t the concept interesting only for holomorphic domains which are not quite complex manifolds?
Not sure what #3 refers to. What I mean in #2 is this remark which says that between complex manifolds, the biholomorphisms are precisely the bijective holomorphisms.
This means that the concept of biholomorphism is interesting only for domains which are not quite complex manifolds (see Bedford’s review).
Hence, I think that your edit announced in #1 needs to be reverted. We should not say in the Idea-section that a biholomorphism is necessarily between complex manifolds.
Urs, re #2, what do you mean by a holomorphic domain?
There is the notion of domain of holomorphy, which is about connected open subsets of $\mathbb{C}^n$ that are domains of holomorphic functions that cannot be extended to a larger such. But maybe you don’t mean that.
In any case, the word “domain” in this context generally suggests to me an open subset of $\mathbb{C}^n$. Open subsets of $\mathbb{C}^n$ are complex manifolds, just not closed complex manifolds.
On the other hand, the first paragraph of Bedford says that a bijective holomorphic mapping between two open sets $D_1, D_2$ of $\mathbb{C}^n$ (not necessarily domains of holomorphy) is automatically a biholomorphism. If I’m not mistaken, this seems to imply that any bijective holomorphic mapping between complex manifolds of the same dimension is a biholomorphism, which confirms something else you said in this thread, except that compactness is not necessary as a hypothesis (as in the remark you linked to). Or am I mistaken?
Edit: Apparently I’m not mistaken. Maybe the MO discussion was meant to point more toward Francesco’s answer than Colin’s.
Right, I shouldn’t have said “holomorphic domain”, sorry. All the more, the entry must not restrict attention to tame spaces.
My answer to the referenced MO question may be wrong. A clarification was seeked in this other MO question. Until the dust settles, I’m deleting the relevant paragraph at the Lab entry on biholomorphic function and replacing it with the Griffiths-Harris/Bedford statement as also articulated by Franceso.
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