$\overline\partial$-closed extension of $CR$-forms with singularities on a~generic manifold

T. N. Nikitina

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 2 (2005), pp. 264-290

Résumé

Let $\Gamma$ be a smooth generic manifold with nonzero Levi form in a domain of holomorphy $\Omega\subset\mathbb C^n$, $n>1$. Let $\Omega_\Gamma\subset\Omega$ be the domain adjacent to $\Gamma$ to which all $CR$-forms defined on $\Gamma$ extend $\overline\partial$-closely. Let $K=\widehat K_\Omega\subset\Omega$ be a holomorphically convex compact set. We show that every $CR$-form on $\Gamma\setminus K$ of bidegree $(l,r)$ with coefficients in $C^1(\Gamma\setminus K)$ extends $\overline\partial$-closely to $\Omega_\Gamma\setminus K$. When $n=2$ and $r=0$ the manifold $\Gamma$ must be closed $(\partial\Gamma=0)$. The proof uses an integral representation, obtained from the integral representation of Airapetyan and Khenkin, in which the integration is carried out over the $CR$-manifold $\Gamma$ only (but not over its complement). In this paper we also consider the problem of $\overline\partial$-closed continuation of $CR$-forms given on $\Gamma\setminus K$, where $\Gamma$ is a generic manifold with nondegenerate Levi form, and $K$ is a meromorphically $p$-convex compactum. We derive some conditions on $\Gamma$, relative to $p$-convexity and $q$-concavity, under which every $CR$-form with smooth coefficients given on $\Gamma\setminus K$ extends $\overline\partial$-closely in some domain $\Omega_\Gamma\setminus K$, where $\Omega_\Gamma$ is a wedge domain with edge $\Gamma$. Our results are local.