ag.algebraic geometry - 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