Cohomologies of rational differential forms of degree n in the space \({\mathbb{C}}\) n
type: Item
instance of: scholarly article
title: Cohomologies of rational differential forms of degree n in the space \({\mathbb{C}}\) n
zbMATH Open document ID: 0654.14011
DOI: 10.1007/BF01139136
author: A. A. Alyakrinskij
author: Avgust K. Tsikh
published in: Mathematical Notes
publication date: 1988-01-01T00:00:00Z
review text: The aim of this paper is to compute explicitly \(H^ n({\mathbb{C}}^ n\setminus V)\) where \(V{\mathbb{C}}^ n\) is an algebraic hypersurface in \({\mathbb{C}}^ n.\) Let \(Q=0\) be the equation of V and \(Q_ q\) the principal part of Q (i.e. the homogeneous part of maximal degree (\(=q)\) of Q). The principal part of Q is called nondegenerate if \(d_ z(Q_ q)\) vanishes only in \(z=0\in {\mathbb{C}}^ n.\)- There are two results: Theorem 1. Suppose V is smooth and the principal part of Q is nondegenerate. Then: \(H^ n({\mathbb{C}}^ n\setminus V)\cong {\mathbb{C}}[z]/\dot {\mathcal I}_ q\) and has dimension \((q-1)^ n\). Moreover if \(\{e_{\beta}\}\) is a monomial base in \({\mathbb{C}}[z]/\dot {\mathcal I}_ q\) then the images of the rational differential forms \(\{e_{\beta}dz_ 1\wedge...\wedge dz_ n/Q\}\) give a base in \(H^ n({\mathbb{C}}^ n\setminus V)\). Here \(\dot {\mathcal I}_ q\) is the jacobian ideal of \(Q_ q.\) Theorem 2. Suppose \(V{\mathbb{C}}^ 2\) is an irreducible curve and the principal part of Q is nondegenerate. Then: \(H^ 2({\mathbb{C}}^ 2\setminus V)\cong {\mathbb{C}}[z]/K_ Q\) where \(K_ Q=\{P\in {\mathbb{C}}[z]| PQ^ 2\) belongs to the jacobian ideal of Q\}. The proof use among other things the Leray exact sequence and the mixed Hodge structure of \(H^{n-1}(V)\).
Mathematics Subject Classification ID: 14F25
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 32B15
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 32C35
zbMATH DE Number: 4067084
zbMATH Keywords: cohomology of complement of algebraic hypersurface
zbMATH Keywords: rational differential forms
zbMATH Keywords: jacobian ideal
zbMATH Keywords: Leray exact sequence
zbMATH Keywords: mixed Hodge structure
reviewed by: Nicolae Mihalache
MaRDI profile type: Publication
cites work: Q3312348
cites work: Q3889369
cites work: Lacunas for hyperbolic differential operators with constant coefficients. II
cites work: Q3846457
cites work: Q5630409
cites work: Integrability in codimension 1
cites work: Q4116546
cites work: Limits of Hodge structures
full work available at URL: https://doi.org/10.1007/bf01139136
OpenAlex ID: W2043211452
Recommended article: Q3737764
Recommended article: Cohomology of coinvariant differential forms
Recommended article: THE DE RHAM COHOMOLOGY OF DIFFERENTIAL SPACES
Recommended article: Cohomology of regular differential forms for affine curves
Recommended article: Cohomology of rational forms and a vanishing theorem on toric varieties
Recommended article: Degrees of cohomology classes of multisingularities in Hurwitz spaces of rational functions
Recommended article: Commutative cochains of germs of differential forms for Euclidean compacta
Recommended article: On the cohomology of regular differential forms and dualizing sheaves
Recommended article: A cohomology for vector valued differential forms
description: scientific article; zbMATH DE number 4067084