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 = {*{20}c} \cup \\ {1 < \left| {i - j} \right| < d - 1} \\ \{ z_i = z_j = 0\} $ Z = \begin{array}{*{20}c} \cup \\ {1 < \left| {i - j} \right| < d - 1} \\ \end{array} \{ z_i = z_j = 0\} in 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 ℂ ^d of the set of planes Z. We explicitly describe cycles that generate groups H _d+2(ℂ ^d \ Z) and H _d−3($ \bar Z $ \bar Z ), where $ \bar Z $ \bar Z = Z ∪ {∞}.