Principal bundle for contractible group is weak homotopy equivalence for ind schemes This is may be obvious, but I am not comfortable with ind-schemes. 
I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular varieties, which is about as smooth as an ind-scheme can be.
I have an unipotent ind-group $U$. More precisely, $U$ is a group object in the category of ind-schemes, and $U$ has a descending filtration all of whose quotients are $\mathbb{G}_a$'s. The group $U$ acts freely on $X$.
There is a quasi-projective variety $Y$, and a map $f: X \to Y$, which is a principal $U$-bundle. (Meaning that there is a cover $Y = \bigcup V_i$ and $f^{-1}(V_i)$ is isomorphic to $U \times V_i$.)

Is $X(\mathbb{C}) \to Y(\mathbb{C})$ a weak homotopy equivalence, using the analytic topology on both sides?

 A: My recollection is that when you turn these into analytic spaces you get something which is locally contractible topologically. In this case what you are describing is a principal bundle for locally contractible spaces in which the fiber is contractible. If the base is paracompact then this will indeed be a weak equivalence, in fact the space corresponding to X will be a topological product space $X = U \times Y$.
This follows because you can build a global section (trivialization). How do you do this? You start with you local trivializations, and you choose a partition of unity subordinate to this cover. You also choose a contraction of U. You can then patch these together to obtain a global section. The exact method and formula is explained, for example, in the appendix of this paper. (This is probably not the only/first/best reference).
Segal, G. Cohomology of topological groups. 1970 Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) pp. 377--387 Academic Press, London
So the real question is whether your space Y is paracompact. I'm pretty sure that your conditions (that Y is quasi-projective) ensure that this is the case. 
