On the global lifting of meromorphic forms
type: Item
instance of: scholarly article
title: On the global lifting of meromorphic forms
zbMATH Open document ID: 0565.32012
DOI: 10.1007/BF01174586
author: Alicia Dickenstein
author: Carmen Sessa
author: Miguel E. Herrera
published in: Manuscripta Mathematica
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155004
review text: Let Y be a complex hypersurface in the complex manifold X. By a ''Weil 1- datum'' we mean a family \({\tilde \omega}=\{\omega_{\alpha}\in \Gamma (U_{\alpha},\Omega^ r(^*Y))\}\) of meromorphic r-forms with poles on \(\tilde Y\) satisfying the following compatability conditions: (1) \(\{\) \(U_{\alpha}\}\) is an open covering of X; (2) (\(\omega\) \({}_{\alpha}- \omega_{\beta})|_{U_{\alpha}\cap U_{\beta}}\) is regular on \(U_{\alpha}\cap U_{\beta}\) forall couples (\(\alpha\),\(\beta)\). A ''global lifting'' of a Weil 1-datum \({\tilde \omega}\) is a form \(\omega \in \Gamma (X,\Omega^ r(^*Y))\) such that for all \(\alpha\), \(\omega - \omega_{\alpha}|_{U_{\alpha}}\) is regular. In this paper, the authors construct the obstruction to the existence of a global lifting. More precisely, we write \(R_ Z: \Gamma (W,\Omega^ r(^*Z))\to \Gamma (W,'D^{r,1})\) for the residue operator associated to Z, where Z is a hypersurface of a complex manifold W. From the basic property of the residue operator, we have \(R_{Y\cap U_{\alpha}}(\omega_{\alpha})=R_{Y\cap U_{\beta}}(\omega_{\beta})\in \Gamma (U_{\alpha \cap U_{\beta}},'D^{r,1})\) for all couples (\(\alpha\),\(\beta)\). Thus we have a global current T of degree (r,1) on X such that \(T|_{U_{\alpha}}=R_{U_{\alpha}\cap Y}(\omega_{\alpha})\). From the basic property of the residue operator, we also know T is \({\bar \partial}\)-closed. Thus T defines a class [\({\tilde \omega}\)] in \(H^ 1(X,\Omega^ r)\). The following theorem is proved: Let the notation be as above. Assume X is projective or Stein. Then there exists a global lifting of the Weil 1-datum \({\tilde \omega}\) if and only if the class [\({\tilde \omega}\)]\(\in H^ 1(X,\Omega^ r)\) is zero. Actually the authors prove the above theorem for the more general setting, namely for the Weil j-datum when Y is the union of hypersurfaces in complete intersection position.
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 32A27
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 3901155
zbMATH Keywords: meromorphic forms
zbMATH Keywords: residue operator
zbMATH Keywords: global lifting of the Weil 1-datum
MaRDI profile type: Publication
cites work: On the De Rham cohomology of algebraic varieties
cites work: Q4195061
cites work: Residues and principal values on complex spaces
cites work: Sur la théorie des formes différentielles attachées à une variété analytique complexe
description: scientific article; zbMATH DE number 3901155