The homology and cohomology of the complements to some arrangements of codimension two complex planes
type: Item
instance of: scholarly article
title: The homology and cohomology of the complements to some arrangements of codimension two complex planes
zbMATH Open document ID: 1232.32016
DOI: 10.1134/S0037446611030189
author: Yu. V. Èliyashev
published in: Siberian Mathematical Journal
publication date: 2011-10-25T00:00:00Z
review text: 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.
reviewed by: Eberhard Oeljeklaus
Mathematics Subject Classification ID: 32Q55
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 52B70
zbMATH DE Number: 5963622
zbMATH Keywords: singular homology
zbMATH Keywords: cohomology
zbMATH Keywords: toric variety
zbMATH Keywords: arrangement of complex planes
zbMATH Keywords: Alexander-Pontryagin duality
zbMATH Keywords: Bochner-Martinelli kernel
MaRDI profile type: Publication
cites work: Q4023363
cites work: Q3346437
cites work: Q4324883
cites work: Recent developments in toric geometry
cites work: Q5711306
cites work: Residual kernels with singularities on coordinate planes
cites work: Topology of plane arrangements and their complements
cites work: Q4537714
cites work: Torus actions and combinatorics of polytopes
cites work: Integral Representations of Holomorphic Functions
cites work: Toric Kempf-Ness sets
cites work: Real quadrics in \(\mathbb C^n\), complex manifolds and convex polytopes
cites work: On the homology groups of arrangements of complex planes of codimension two
full work available at URL: s0037446611030189
OpenAlex ID: W2100193148
description: scientific article; zbMATH DE number 5963622