Normal crossing properties of complex hypersurfaces via logarithmic residues
type: Item
instance of: scholarly article
title: Normal crossing properties of complex hypersurfaces via logarithmic residues
zbMATH Open document ID: 1314.32043
DOI: 10.1112/S0010437X13007860
author: Mathias Schulze
author: Michel Granger
published in: Compositio Mathematica
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.2612
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32A27
Mathematics Subject Classification ID: 32S10
Mathematics Subject Classification ID: 32C37
zbMATH DE Number: 6363203
zbMATH Keywords: normal crossing divisors
zbMATH Keywords: Jacobian ideal
zbMATH Keywords: logarithmic residue
zbMATH Keywords: local duality
zbMATH Keywords: free divisors
zbMATH Keywords: weakly holomorphic functions
MaRDI profile type: Publication
arXiv ID: 1109.2612
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review text: Let \(D\) be a reduced divisor in a complex manifold \(S\). In 1980 K. Saito has noted that if \(D\) is normal crossing outside an analytic subset of codimension at least two in \(D\), then the residue of any logarithmic differential 1-form along \(D\) is holomorphic on the normalization \({\widetilde{D}}\), that is, \(\text{res\,} (\Omega^1_S(\log D)) = {\mathfrak n}_*({\mathcal O}_{\widetilde{D}})\), where \({\mathfrak n} : \widetilde{D} \rightarrow D\) is the morphism of normalization (see [\textit{K. Saito}, J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265--291 (1980; Zbl 0496.32007)], Lemma \(2.13\)). There he has also stated (Theorem~\(2.11\)) that the converse is true for plane curves. The paper under review contains a proof of the corresponding statement for divisors of arbitrary dimension. For this, the authors introduce a dual logarithmic residue map and then use nice relations between properties of the Jacobian ideal and the normalization studied also in detail by \textit{E. Faber} [Math. Ann. 361, No. 3--4, 995--1020 (2015; Zbl 1333.32034)]. In conclusion, some interesting applications for free divisors are discussed.
reviewed by: Aleksandr G. Aleksandrov
description: scientific article; zbMATH DE number 6363203