Let $M$ denote a complex manifold of dimension $n$ and let $X\subset M$ denote an analytic hypersurface. Then it is a standard fact from several complex variables that around a given point $p\in X$ there are open subsets $V\subset X, W\subset \mathbb{C}^{n-1}$ and and a finite-sheeted covering $\pi: V\rightarrow W$ branched over an analytic set $A$. Frequently, $W$ is taken to be a polydisc (see, for example, Griffiths & Harris). Now set $V' = V\setminus \pi^{-1}(A)$ and $W' = W\setminus A$. If $V'$ is connected, then $\pi$ induces a genuine (holomorphic) covering $\pi':V'\rightarrow W'$.

For appropriately chosen basepoints $v_0, w_0$ in $V',W'$, respectively, covering space theory says that $\pi'$ induces an injection $\pi_1(V',v_0)\rightarrow \pi_1(W',w_0)$, and in some cases (e.g. involving curves) this might be enough to figure out $\pi_1(V',v_0)$ if $\pi_1(W',w_0)$ is known. If $V$ were smooth, $\pi^{-1}(A)$ would be an analytic subset of complex codimension 1 and the inclusion $V'\hookrightarrow V$ would induce a surjection $\pi_1(V',v_0)\rightarrow \pi_1(V,v_0)$, determining a presentation for $\pi_1(V,v_0)$. However, $V$ may not be smooth, and I am not sure in what generality this map is still a surjection, although clearly I'd like to know.

What I really want to do is determine a presentation for $\pi_1(X,x_0)$. It seems that one way to to go about doing this would be to construct an open cover $(V_{\alpha})$ of $X$ consisting of open sets with the same properties possessed by $V$ above AND such that $(V_{\alpha})$ satisfies the hypotheses of the Seifert-van Kampen Theorem (see, for example, Hatcher). In the very best case, one could arrange for the intersections $V_{\alpha}\cap V_{\beta}$ to be simply-connected and for the triple intersections to be path-connected. Then one could read off a presentation of $\pi_1(X,x_0)$ in terms of the presentations of the $\pi_1(V_{\alpha}, v_{0\alpha})$.

I'd be interested to know when such a cover $(V_{\alpha})$ exists, especially for $n\geq 3$.

I am also interested in hearing about any known methods of calculating the fundamental group of an analytic hypersurface (not necessarily smooth), in general or in special cases (for example, the Lefschetz Hyperplane Theorem can be used on certain projective hypersurfaces).

(Note: I'd like to consult Dimca's book "Singularities and Topology of Hypersurfaces", but I'll be away from my library for the next few weeks.)

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