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• CommentRowNumber1.
• CommentAuthorColin Tan
• CommentTimeJul 19th 2014

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?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 19th 2014

Isn’t the concept interesting only for holomorphic domains which are not quite complex manifolds?

• CommentRowNumber3.
• CommentAuthorColin Tan
• CommentTimeJul 19th 2014
To prevent evil, we would need to be able to say that any two complex projective spaces of the same dimension are biholomorphic.
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 19th 2014

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.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJul 19th 2014
• (edited Jul 19th 2014)

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.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 19th 2014

Right, I shouldn’t have said “holomorphic domain”, sorry. All the more, the entry must not restrict attention to tame spaces.

• CommentRowNumber7.
• CommentAuthorColin Tan
• CommentTimeJul 20th 2014

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.

• CommentRowNumber8.
• CommentAuthorColin Tan
• CommentTimeJul 20th 2014
• (edited Jul 20th 2014)
Also reverting back the definition till the original.

I apologize for any errors I have made and the confusion that I have caused.