On the homology of subvarieties of toric varieties
type: Item
instance of: scholarly article
title: On the homology of subvarieties of toric varieties
zbMATH Open document ID: 1208.14016
DOI: 10.1134/S1064562410010308
author: N. A. Buruchenko
published in: Doklady Mathematics
publication date: 2010-11-17T00:00:00Z
review text: In this paper the author considers a quasiaffine variety \(Y\backslash V\) represented as the difference of two algebraic varieties in \(\mathbb C^n\). The structure of the compact homology groups \(H_p(Y\backslash V)\) is described by Froissart's theorem [\textit{T. Fotiadi, M. Froissart, J. Lascoux} and \textit{F. Pham}, Topology 4, 159--191 (1965; Zbl 0173.09301)], which asserts that if \(V_i=S_i\cap Y\), the family of hypersurfaces \(S\) in \(\mathbb P^n\) is generic and \(V=\bigcup V_i\), then \[ H_p(Y\backslash V)=H_p(Y) \oplus [\bigoplus_i \delta H_{p-1}(V_i)] \oplus [\bigoplus_{i,j} \delta^2 H_{p-2}(V_i \cap V_j)] \oplus \dots, \] where the direct sum on the right is taken over all possible intersections of the family \({V_i}_i\) and \(\delta, \delta^2,\dots\) are iterated coboundary (tube) Leray operators [\textit{J. Leray}, Bull. Soc. Math. Fr. 87, 81--180 (1959; Zbl 0199.41203)]. The first theorem in the paper explains the reason of the decomposition of \(H_p(Y\backslash V)\) above. Indeed, they show a generalization of Froissart's theorem for compact toric \(n\)-dimensional spaces. Quasitoric varieties are defined as the difference \(Y\backslash V\) of two analytic subvarieties in some compact complex toric space \(X\). It turns out that the quasitoric variety \(Y\backslash V\) being generic is insufficient for the validity of a decomposition of \(H_p(Y\backslash V)\) as the one before. They show that for the validity of the decomposition it is necessary to impose additional constraints on the rational equivalence of cycles. Precisely, in Theorem 2 they give conditions on the class of cycles of \(X\) for which a decomposition \(H_p(Y\backslash V)=H_p(Y) \oplus [\bigoplus_i \delta H_{p-1}(V_i)] \oplus\dots\) holds. As a corollary, they conclude that for a generic algebraic hypersurface \(F\) in the complex torus \(\mathbb T^n\), \(H_p(F,\mathbb Z)\cong \mathbb Z^{n\choose p}\) for all \(p=0,\dots,n-2\).
reviewed by: Nicolás Botbol
Mathematics Subject Classification ID: 14F25
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 5816874
zbMATH Keywords: quasiaffine variety
zbMATH Keywords: compact homology
zbMATH Keywords: Froissart's theorem
zbMATH Keywords: quasitoric varieties
zbMATH Keywords: toric space
zbMATH Keywords: toric variety
MaRDI profile type: Publication
cites work: Le calcul différentiel et intégral sur une variété analytique complexe. (Problème de Cauchy. III.)
cites work: Q4023363
cites work: Residual kernels with singularities on coordinate planes
description: scientific article; zbMATH DE number 5816874