Residues of holomorphic maps between manifolds
type: Item
instance of: scholarly article
title: Residues of holomorphic maps between manifolds
zbMATH Open document ID: 0962.32023
author: El Hadji Cheikh Mbacké Diop
published in: Hokkaido Mathematical Journal
publication date: 2000-11-22T00:00:00Z
review text: Let \(f: V\to W\) be a holomorphic map between complex analytic manifolds \(V\) and \(W\) respectively of dimensions \(n\) and \(q\), \(n\geq q.\) Assume that the singular set \(\Sigma\) of \(f\) is not empty. Then the Chern classes \(c_{n-q+i} [TV -f^{-1}(TW)]\) of the virtual bundle \([TV -f^{-1}(TW)]\) localize near \(\Sigma\) for \(i = 1,\dots,q.\) If \(\Sigma\) is compact, then the residue of \(c_{n-q+i} [TV -f^{-1}(TW)]\) is defined in the homology group \(H_{2q-2i}(\Sigma)\). The author computes these residues assuming the singular fibres of \(f\) are locally isolated complete intersections and the singular set \(\Sigma\) is smooth. If \(q = 1\) and \(V\) is compact, the Fulton formula computes the cohomology class \([c_n(V) - (c_{n-1}(V) \cup f^\ast_{c_1}(W))]\cap[V]\) using the Milnor number. The main result of the paper is a generalization of this formula for all codimension \(q.\) A generalization of the formulas of Riemann-Hurwitz type is deduced from it. Two examples, for which the residues are not equal to zero, are considered.
Mathematics Subject Classification ID: 32S20
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 1448535
zbMATH Keywords: residues of holomorphic maps
zbMATH Keywords: characteristic classes and numbers
zbMATH Keywords: Chern classes
zbMATH Keywords: virtual bundle
zbMATH Keywords: residue
zbMATH Keywords: homology group
zbMATH Keywords: singular fibres
zbMATH Keywords: Milnor number
zbMATH Keywords: intersections
zbMATH Keywords: singular set
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350912963
OpenAlex ID: W2103162734
DOI: 10.14492/HOKMJ/1350912963
description: scientific article; zbMATH DE number 1448535