# Lifting analytic map

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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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.
So according to the notations (I believe they are standard), this fiber product should be denoted as $\Delta\times _{V/G} V$.
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