Let $n\geq2$. Let $G$ be a linear automorphisms group of prime order on $\mathbb{C}^n$. We assume that $0$ is the unique fixed point of $G$. I consider the quotient $\mathbb{C}^n/G$. It is a toric variety, so I can consider the toric blowup: $\widetilde{\mathbb{C}^{n}/G}$. I am interesting in the integral cohomology (singular cohomology) of $\widetilde{\mathbb{C}^{n}/G}$. Especially, I would like to prove that $H^{2k}(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ is torsion free for $k\leq n-1$. Do someone know an efficient method to deal with this kind of problems? Has this cohomology been studied somewhere?
I managed to prove that $H^2(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ and $H^{2n-2}(\widetilde{\mathbb{C}^{n}/G},\mathbb{Z})$ are torsion free using Danilov combinatorial result. However for the other cohomology groups, it becomes too technical to be done by hand.
