I wonder if there is any situation where one can talk about Gromov-Witten invariants or quantum multiplication for singular varieties. Ideally, I would like have a situation where for a singular variety $X$ one can define quantum multiplication operators by elements of ORDINARY cohomology of $X$ on the INTERSECTION cohomology of $X$ (I have some examples where I know what I want the answer to be, but I don't know how to ask the question).
In fact, I will be ready to start with the following simple example: assume that $X$ just has quotient singularities, i.e. locally it looks like $Y/G$ where $Y$ is smooth and $G$ is a finite group. In this case the intersection cohomology coincides with the ordinary cohomology, so my question is whether in this case one can define quantum multiplication. One warning: I am talking about quantum cohomology of $X$ itself, not about what is called "orbifold quantum cohomology" (which in many cases coincides with the quantum cohomology of a good resolution of $X$).
