## Cohomology of toric blowup

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.