Can the fundamental group of a quasi-affine variety over $\mathbb{C}$ be a torsion group?



The fundamental group of a complex, quasi-affine variety
The answer is *yes*, already for an affine variety.


The following example is taken from Dimca's book *Singularities and topology of hypersurfaces*, see page 102 and page 105. We work over $\mathbb{C}$.


Let $V \subset \mathbb{P}^n$ be a hypersurface and $U:=\mathbb{P}^n \setminus V$ its complement. Since $V$ is very ample, $U$ is an affine variety. Then we have the following



> 
> **Proposition.** Assume that $V$ has $k$ irreducible components $V\_1, \ldots, V\_k$ with $\deg V\_i =d\_i$. Then $$H\_1(U, \mathbb{Z})=\mathbb{Z}^{k-1} \oplus \mathbb{Z}/d \mathbb{Z},$$
>  where $d$ denotes the greatest common divisor of the integers $d\_1, \ldots, d\_k$.
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> If moreover $V$ has only normal crossing singularities in codimension $1$, then $\pi\_1(U)$ is abelian, hence it is isomorphic to the group $H\_1(U, \mathbb{Z})$ given above.
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> In particular, if $V$ is irreducible and normal of degree $d$, one has $$\pi\_1(U)=\mathbb{Z}/d \mathbb{Z}.$$
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> 


**Remark.** If $V$ is irreducible of degree $d$, then the Proposition above implies that in any case the abelianization of $\pi\_1(U)$ is isomorphic to $\mathbb{Z}/d \mathbb{Z}$. However, when $V$ is not normal crossing in codimension $1$, the group $\pi\_1(U)$ may be nonabelian (and also infinite). Let me give a couple of classical examples, that can be found in Dimca's book, Chapter 4.


*Example 1.* Let $V \subset \mathbb{P}^2$ be the tricuspidal quartic curve of equation $$x^2y^2+y^2z^2+z^2x^2-2xyz(x+y+z)=0.$$ Then $\pi\_1(U)$ is isomorphic to the metacyclic group of order $12$. This is a finite nonabelian group, whose abelianization is $\mathbb{Z}/4 \mathbb{Z}$.


*Example 2.* Let $V \subset \mathbb{P}^2$ be the plane sextic of equation $$(x^2+y^2)^3+(y^3+z^3)^2=0.$$ $V$ has six cusps situated on a conic and $\pi\_1(U)=(\mathbb{Z}/2 \mathbb{Z}) \ast (\mathbb{Z}/3 \mathbb{Z})$. This is an infinite, nonabelian group whose abelianization is $(\mathbb{Z}/2 \mathbb{Z}) \oplus (\mathbb{Z}/3 \mathbb{Z})= \mathbb{Z}/6 \mathbb{Z}$.


