Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold).



> 
> Is $SL(2,\mathbb C)/SL(2,\mathbb Z)$ a quasi-projective variety?
> 
> 
> 


The natural generalization of this question seems to be the following. Let $G$ be a semisimple linear algebraic group over $\mathbb Q$. Then $G(\mathbb Z)$ is well-defined up to taking a finite-index subgroup. Thus we can ask the same question: is the complex manifold $G(\mathbb C)/G(\mathbb Z)$ a quasi-projective variety?



Is SL(2,C)/SL(2,Z) a quasi-projective variety?
No, the quotient is not quasi-projective. In the paper [Invariant meromorphic functions on complex semisimple Lie groups](http://digreg.mathguide.de/cgi-bin/ssgfi/anzeige.pl?db=reg&ti=I%2a&nr=092504&ew=SSGFI) by D. N. Ahiezer you can find the following result.



> 
> **Theorem.** Let $G$ be a connected semisimple linear algebraic group
>  defined over the rationals and
>  $\Gamma$ be a subgroup of $G(\mathbb
> > Q)$ which is Zariski dense in $G$.
>  Then there are no invariant analytic
>  hypersurfaces in $G$ invariant by the
>  action of $\Gamma$. In particular, if
>  the quotient $G/ \Gamma$ exists as a
>  complex variety then every 
>  meromorphic function on it is constant.
> 
> 
> 


This implies that the quotient is not quasi-projective and even more: it cannot be holomorphically embedded in any algebraic variety. 


