Complex moduli space (or Teichmuller space) of a Quintic Calabi-Yau 3-fold is a 
101-dimensional complex orbifold. Does it have a toric structure? 



Is the complex moduli of Quintic Calabi-Yau toric?
The complex moduli space does not admit a toric strucutre, since the orbifold fundamental group of a toric orbifold must be abelian. Indeed, $\pi\_1(\mathbb C^\*)^n$ surjects on the orbifold fundamental group. Also, the orbifold stabisier of each point on a toric orbifold 
is a finite abelian group. At the same time the stabiliser of the quintic $\sum\_i z^5=0$
is a non-comutative group. Also I am sure that the orbifold fundamental group of the moduli space of quintics contains free (non-abelian) subgroups, but I don't know how to prove it. 


Also it should be true that the Tiechmuller space is not algebraic. It least this happen in lower dimensions for cubics in $\mathbb CP^2$ and for quartics in $\mathbb CP^3$.
In the first case the Theichmuiller space is a disk, and in the second it is  

a hermitian domain of type IV. Moduli spaces of polarised K3 are discussed here
for example, here:


<http://people.bath.ac.uk/masgks/Papers/k3moduli.pdf>


