Boundary conditions for a Poisson equation on the resolution of an orbifold singularity

Question

I'm looking to solve a Poisson equation on a curved space constructed from the following paper: https://arxiv.org/abs/hep-th/0701227v3, specifically for the n=2 case. The space in consideration is the blow-up of singularities from a $T^4 / \mathbb{Z}_2$ orbifold. Unfortunately, I don't know enough toric/algebraic geometry to understand how the blow-up was constructed.

Does anyone know how I can find the appropriate boundary conditions in such a case? Do they arise as part of the resolution, or can they be inferred physically somehow? I understand that in the blow-down limit, we regain the orbifold which inherently has periodic boundary conditions as well as a $\mathbb{Z}_2$ identification, but I'm skeptical that these still apply in the blow-up, especially since the space is now defined locally around each singularity.

Any help would be appreciated.