Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ is Gorenstein ($K\_X$ is Cartier) and $R^1\pi\_\* \mathcal{O}\_Y=\mathbb{C}\_x$ where $\pi:Y\to X$ is a resolution of the singularity.


Equivalently: $\pi\_\*\omega\_Y=m\_x\omega\_X$.


Thanks in advance,
Benoit



local fundamental group of elliptic singularities
Well it is not always the case: elliptic singularities of surfaces can have very complicated local fundamental group. For instance, the following example
$$x^2+y^3+z^7=0$$
has a finite index subgroup which the fundamental group of a curve of genus 3 curve. To see this, note that it can be realized as a finite quotient of the cone over the Kein quartic $C$. This curve has the biggest automorphism group among curves of genus 3 ($\mathrm{Aut}(C)$ has order 168) and it acts naturally on the cone over $C$. This description can be found in some old papers of Dolgachev "Conic quotient singularities of complex surfaces" 
<http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=58860&vfpref=html&r=72&mx-pid=345974>
(see also some unpublished paper of Miles Reid).


