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?
> 
> 
> 



Principal bundle for contractible group is weak homotopy equivalence for ind schemes
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. 


