TITLE: Topological fundamental group of a variety
QUESTION [8 upvotes]: I have an explicit question. 
I have a complex projective variety defined by $2\times 2$ minors of a matrix. The entries are polynomials from a weighted projective space. In fact, it's a singular 3-fold with only quotient singularities. 
Now I want to show that it is simply connected. 
Can anyone help in giving direction in this regard?

REPLY [3 votes]: I am not sure this will help, but the methods we use in Fundamental Group of  Moduli Spaces of Representations might be useful (if you are working over $\mathbb{C}$).
In particular, these two properties are worth pointing out:

Given any non-empty Zariski open subset $U \subset V$, where $V$ is normal, the homomorphism $\pi_1(U) \to \pi_1(V)$ induced by the inclusion map is surjective.
The GIT quotient map $V\to V//G$ is $\pi_1$-surjective, where $G$ is a connected reductive affine algebraic group.

Another tool that might be of interest is Proposition 5.8 which gives three conditions for the inclusion map $V\setminus W\hookrightarrow V$ to be 2-connected (where $W\subset V$ is a subvariety).