Let X be an algebraic variety over $\mathbb C$. 1. Is it possible to compute its fundamental group? 2. If X is two dimensional, what is its fundamental group? 3. Let $X\to \bar X$ be the inclusion to its compactification, and suppose $\bar X-X$ is a normal crossing singularity. What is the relation of the two fundamental groups?
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2$\begingroup$ What do you mean with 'compute'? There are algorithms giving you a triangulation of an algebraic variety, and from that giving a presentation of the fundamental group is trivial, but is this what you mean? $\endgroup$– Denis NardinCommented Jun 3, 2019 at 20:39
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$\begingroup$ Yes. I would like to know the exact presentation of the fundamental group. For example, is it a free group? Or a finitely presented group and what are the relations? Is it possible for general varieties? Any reference to understand the algorithm? $\endgroup$– LongmaCommented Jun 4, 2019 at 7:50
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$\begingroup$ Note that having a presentation does not imply being able to answer your questions—not even, in full generality, whether the presented group is trivial. $\endgroup$– LSpiceCommented Jun 4, 2019 at 14:43
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