On the homology groups of arrangements of complex planes of codimension two

In the study of two-dimensional compact toric varieties, there naturally appears a set of coordinate planes of codimension two $Z=\cup_{1<|i-j|<d-1}\{z_i=z_j=0\}$ in $\mathbb C^d$. Based on the Alexander–Pontryagin duality theory, we construct a cycle that is dual to the generator of the highest dimensional nontrivial homology group of the complement in $\mathbb C^d$ of the set of planes $Z$. We explicitly describe cycles that generate groups $H_{d+2}(\mathbb C^d\setminus Z)$ and $H_{d-3}(\overline Z)$, where $\overline Z=Z\cup\{\infty\}$.