# How to study the nonregular part of a finite branched holomorphic covering?

A finite branched holomorphic covering is a holomorphic map $f : V \to W$ between holomorphic varieties $V$ and $W$ such that

• $f$ is a finite branched covering (in the topological sense)
• There is a regular part $V_0 \subset V$ on which $f$ is a locally biholomorphic mapping

On the regular part, the behavior of $f$ is relatively simple since it is locally biholomorphic, and locally $V_0$ looks just like $W_0 = f(V_0)$. My question is: do we (can we) know anything about the nonregular part (maybe that's not the right term)? I.e., can we know the structure of the map $f:(V \setminus V_0) \to W$?

In particular I am interested in the case where $W$ is simply a open disc of $\mathbb{C}$ centered at the origin, and $W_0$ is the whole disk without the origin. How should I study the structure of $f^{-1}(0)$?

This is only a partial answer. Regarding the last part of your question:

Every (connected) finite branched cover $$f: V\rightarrow \mathbb{D}$$ of the disc is isomorphic to one of the form $$p_n: \mathbb{D}\rightarrow \mathbb{D}$$, where $$p_n(z) = z^n$$ and $$n\in \mathbb{Z}^+$$. In particular, $$f^{-1}(0)$$ consists of a single point since $$z^n$$ has a unique zero.

To get a better idea of why only the map $$z\rightarrow z^n$$ appears, note that every such branched covering arises from an unbranched covering $$g:X\rightarrow \mathbb{D}^*$$ which one can also show is of the form $$p_k: \mathbb{D}^* \rightarrow \mathbb{D}^*$$ up to isomorphism. The original branched covering $$V\rightarrow \mathbb{D}$$ is obtained by completing, i.e. "adding an extra point" to $$\mathbb{D}^*$$ and extending $$g$$ so that it takes the value $$0$$ on this new point. So $$f^{-1}(0)$$ consists of a single point (of "multiplicity" $$k$$) by construction. Now, if you don't require $$V$$ to be connected, you'll end up with possibly several disjoint copies of this situation, namely $$N$$ discs $$\mathbb{D}_j$$ equipped with maps $$z\rightarrow z^{n_j}$$. Then $$f^{-1}(0)$$ consists of $$N$$ points $$v_j$$ with multiplicities $$n_j$$. This is the situation you'll find when analyzing the preimage of a small embedded disc under a non-constant map of Riemann surfaces.

In higher dimensions, branched coverings are more difficult to understand. One useful fact is that such maps are proper (by definition). So if $$V,W$$ are smooth, then the Proper Mapping Theorem applies. In particular, $$f(V\backslash V_0)$$ is an analytic hypersurface in $$W$$ (since the ramification divisor $$V\backslash V_0$$ is an analytic hypersurface in $$V$$ - see Griffiths & Harris).

Edit: One nice reference for the basics of branched covers is the book "Holomorphic Functions to Complex Manifolds" by Grauert and Fritzsche.

• Thank you so much for the answer. For the first part, what's the name of the theorem? I have heard different names like "local uniformization theorem" and "local canonical form", and I had trouble finding the original statement and proof. Why $f^{-1}(0)$ is just a single point: why can't a few sheets coalesce into one point in $f^{-1}(0)$ while the remaining sheets coalesce into a different point? Jul 5, 2012 at 15:49
• Hi ssquidd. I was going to reply with a comment, but it ran too long. I've edited the my answer to reflect your question. Jul 5, 2012 at 17:48
• Miranda calls the local expression $z\rightarrow z^n$ of a holomorphic map between Riemann surfaces the "local normal form". That such maps have a this local form follows easily from power series manipulations. Check out his book "Algebraic Curves and Riemann Surfaces" for more. Jul 6, 2012 at 5:16
• So, did you want to know more about the higher dimensional situation or is my answer satisfactory? Jul 20, 2012 at 18:31