The homology and cohomology of the complements to some arrangements of codimension two complex planes
Motivated by problems from the theory of toric varieties the author is interested in the structure of the singular homology groups \(H_s(\mathbb C^d\backslash Z)\) and the de Rham cohomology groups \(H^s(\mathbb C^d\backslash Z), 1\leq s\leq d+2\), with \[ Z=\bigcup_{1<|i-j|<d-1}\big\{(z_1,\dots,z_d)\in\mathbb C^d\,\big|\,z_i=z_j=0\big\}, \] a specific arrangement of twocodimensional coordinate subspaces. For \(1\leq s\leq d+2\) he gives an explicit construction and a geometric description of a system of cycles which constitute a basis for the torsion-free part of \(H_s(\mathbb C^d\backslash Z)\), and for \(1\leq s < d+2\) he also calculates the differential forms which yield the dual basis of \(H^s(\mathbb C^d\backslash Z)\). The cycles can be realized for \(3\leq s\leq d-1\) as products \(S^3\times (S^1)^{s-3}\) of the threedimensional sphere and the \((s-3)\)-dimensional torus. Let \(\omega_{BM}\) denote the Bochner-Martinelli kernel of \(\mathbb C^2\backslash \{0\}\). The differential forms in the dual basis are received, up to a suitable permutation of coordinates, from the forms \[ \omega_{BM}(z_1\cdot \dots\cdot z_k, z_{k+1}\cdot\dots\cdot z_m)\wedge\frac{dz_2}{z_2}\wedge\dots\wedge\frac{dz_{m}}{z_{m}} \] (cf. [\textit{G. Sorani}, Am. J. Math. 88, 737--746 (1966; Zbl 0154.33203)]). The Alexander-Pontryagin duality and results of [\textit{V. M. Buchstaber} and \textit{T. E. Panov}, Torus actions and their applications in topology and combinatorics. University Lecture Series. 24. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1012.52021)] are used for an explicit construction of a basis of \(H_{2d-s-1}(\overline{Z})\), where \(\overline{Z}\) is the compactification of \(Z\) in the one-point compactification \(S^{2d}\) of \(\mathbb C^d\). The highest non-trival cohomology group \(H^{d+2}(\mathbb C^d\backslash Z)\) is cyclic and a generating \((d,2)\)-form for it is explicitly constructed.
scientific article; zbMATH DE number 5963622