Topology of complete intersections
type: Item
instance of: scholarly article
title: Topology of complete intersections
zbMATH Open document ID: 0896.14028
published in: Commentarii Mathematici Helvetici
publication date: 1998-09-28T00:00:00Z
review text: The author considers complete intersections in complex projective spaces. It is a classical result of \textit{R. Thom} that the topology of an \(n\)-dimensional complete intersection depends only on the degrees of the homogeneous polynomials. For an unordered set \({\mathbf d}= (d_1,\dots, d_r)\) of nonnegative integers, \(X_n({\mathbf d})\) will denote an \(n\)-dimensional complete intersection defined by \(r\) homogeneous polynomials in \(n=r+1\) variables and degrees \(d_1,\dots, d_r\), respectively. The total degree of \(X_n({\mathbf d})\) is defined as \(d= d_1d_2\cdots d_r\). The following stems from the author's abstract: Let \(X_n({\mathbf d})\) and \(X_n({\mathbf d}')\) be two \(n\)-dimensional complete intersections with the same total degree \(d\). If \(d\) has no prime factors less than \(\frac{n+3}{2}\) with \(n\) even, then \(X_n({\mathbf d})\) and \(X_n({\mathbf d}')\) are homotopy equivalent if and only if they have same Euler characteristics and signatures. This confirms a conjecture of Libgober and Wood. Furthermore, if \(d\) has no prime factors less than \(\frac{n+3}{2}\), then \(X_n({\mathbf d})\) and \(X_n({\mathbf d}')\) are homeomorphic if and only if their Pontryagin classes and Euler characteristics agree.
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 1121241
zbMATH Keywords: complete intersections
zbMATH Keywords: Euler characteristics
zbMATH Keywords: signatures
zbMATH Keywords: Pontryagin classes
author: Fuquan Fang
reviewed by: Satyagopal Mandal
MaRDI profile type: Publication
full work available at URL: s000140050028
OpenAlex ID: W1983339146
DOI: 10.1007/S000140050028
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description: scientific article; zbMATH DE number 1121241