On the global lifting of meromorphic forms
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.
scientific article; zbMATH DE number 3901155