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

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Abstract

In the study of two-dimensional compact toric varieties, there naturally appears a set of coordinate planes of codimension two \( Z = \begin{array}{*{20}c} \cup \\ {1 < \left| {i - j} \right| < d - 1} \\ \end{array} \{ z_i = z_j = 0\} \) in ℂd. 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 ℂd of the set of planes Z. We explicitly describe cycles that generate groups H d+2(ℂd \ Z) and H d−3(\( \bar Z \) ), where \( \bar Z \) = Z ∪ {∞}.

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Authors and Affiliations

  1. University of Massachusetts, 710 North Pleasant Street, Amherst, MA, 01003, USA

    A. V. Kazanova

  2. Siberian Federal University, pr. Svobodnyi 79, Krasnoyarsk, 660041, Russia

    Yu. V. Eliyashev

Authors
  1. A. V. Kazanova
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  2. Yu. V. Eliyashev
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Correspondence to A. V. Kazanova.

Additional information

Original Russian Text © A. V. Kazanova and Yu. V. Eliyashev, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 10, pp. 33–39.

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Kazanova, A.V., Eliyashev, Y.V. On the homology groups of arrangements of complex planes of codimension two. Russ Math. 53, 28–33 (2009). https://doi.org/10.3103/S1066369X09100041

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  • DOI: https://doi.org/10.3103/S1066369X09100041

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