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I believe the following statement is true:

Given a complex analytic map $f:\Delta\to V/G$, where $\Delta$ is a disc in $\mathbb{C}$, $V$ a finite dimensional complex vector space and $G$ a finite subgroup of $GL(V)$, then $f$ admits an analytic lift $\tilde f:\Delta'\to V$ up to a ramified cover. More precisely, there exists a ramified cover $r:\Delta'\to\Delta$ such that $f\circ r = \pi\circ \tilde f$ where $\pi:V\to V/G$ is the canonical projection.

I think I have a relatively elementary proof. However, this statement sounds very much like a "classical" result, but I have been unable to find a reference. Does anyone knows to whom I should attribute this result ?

Thanks a lot

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up vote 4 down vote accepted

Salut Yann!

You can refer to the general notion of fiber product, for example. This is a construction that works in the complete generality. It might happen that in the situation you consider the fiber product will not be irreducible, (this happen when the preimage of $f(\Delta)$ in $V$ is not irreducible), then you just take one of the irreducible components of this fiber product.

Here is a reference for fiber products : http://planetmath.org/fibreproduct .
So according to the notations (I believe they are standard), this fiber product should be denoted as $\Delta\times _{V/G} V$.

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Hi Dima, can you be somewhat more specific about which fiber product you are looking at ? I guess you are talking about $\Delta\times_f V \to \Delta$ ? –  Yann Jun 5 '11 at 11:52
    
Thanks Dima, this leads to a nice proof. Slicker, although equivalent to the one I had at hand. –  Yann Jun 8 '11 at 9:17

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