The Group Cohomology of Peroidized Hypertoric Variety

To a graph $\Gamma$, one can associate a hypertoric variety $\mathcal{M}(\Gamma)$ and its multiplicative version $\mathcal{M}^{\mathrm{mul}}(\Gamma)$. It was shown in [DMS24] that the cohomology of $\mathcal{M}^{\mathrm{mul}}(\Gamma)$ is computed by the CKS complex, which is a finite dimensional complex attached to $\Gamma$. The multiplicative hypertoric variety can be realized as the quotient of a periodized hypertoric variety by a lattice action. In this paper, we show that the group cohomology of the lattice with coefficients in the cohomology of the prequotient is isomorphic to the cohomology of the CKS complex using a spectral sequence argument. Therefore, the group cohomology can serve as an alternative way to compute the cohomology of multiplicative hypertoric varieties. We also found graph-theoretic descriptions for the Euler characteristics of the graded pieces in a certain decomposition of $\mathrm{H}^\bullet(\mathcal{M}^{\mathrm{mul}}(\Gamma))$.