On the homology groups of arrangements of complex planes of codimension two (Q736172)

In the study of two-dimensional compact toric varieties, there naturally appears a set of coordinate planes of codimension two \( Z = \bigcup_{1 < \left| {i - j} \right| < d - 1} \{ z_i = z_j = 0\} \) 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 \(H_{d-3}(\bar Z)\), where \(\bar Z= Z \cup\{\infty\}\).