Extension d'objets CR. (Extension of CR objects)
The author considers the following situation: Let \(\Sigma\) be a real hypersurface of an open domain \(U\subset {\mathbb{C}}^ n\), \(n\geq 2\), such that \(\Sigma =\{\phi =0\}\) for \(\phi \in C^{\infty}(U)\); let \(M=\{h_ 1=0\}\) for \(h_ 1:U\to {\mathbb{R}}\) pluriharmonic; assume: 1) \(M\cap \Sigma\) is a compact connected submanifold 2) \(M\cap \Sigma\) is the boundary of a bounded domain \(A\subset M\) and of a bounded domain \(S\subset \Sigma\) such that \(S\subset \{h_ 1>0\}.\) Let \(D\subset {\mathbb{C}}^ n\) such that \(bD=\overline{S\cup A}\); let finally \(h_ 2:U\to {\mathbb{R}}\) be another pluriharmonic function such that: i) \(\{\) \(z\in U|\) \(h_ 1(z)=h_ 2(z)=0\}\) is a complex subvariety of dimension n-1. ii) \(D'=\{z\in D|\) \(h_ 2(z)>0\}\) is a domain. D' is called a sector and S is the non-flat component of bD; if \(\phi\) is strictly plurisubharmonic (resp. plurisubharmonic) D' is called strictly pseudoconvex (resp. pseudoconvex). The main results of the paper are the following: Theorem 1. Assume D' is a strictly pseudoconvex sector, then every CR- divisor on S extends to D'. Theorem 2. Assume: i) \(n\geq 5\) and D' is pseudoconvex. ii) There exists a sequence \(\{\epsilon_{\nu}\}\searrow_ 0\) such that \(H^ 3(\{z\in S|\) \(h_ 2(z)>\epsilon_{\nu}\},{\mathbb{Z}})=0\). Then every CR vector bundle of rank 1 on S is the restriction of a unique vector bundle of rank 1, holomorphic on D' and \(C^{\infty}\) on D'\(\cup S.\) Theorem 3. Let \(\Omega \subset {\mathbb{C}}^ n\), \(n\geq 4\), be a bounded pseudoconvex domain such that: i) \(\Omega\) has a fundamental system of Stein neighborhoods, ii) \(H^ 3(b\Omega,{\mathbb{Z}})=0\). Then every CR vector bundle of rank 1 on \(b\Omega\) is the restriction of a vector bundle of rank 1 which is holomorphic on \(\Omega\) and \(C^{\infty}\) on \({\bar \Omega}\).
scientific article; zbMATH DE number 4023694