Boundaries of analytic sets
Presented by Luca BARACCO
Type: Oral presentation
Track: Several Complex Variables
We show that a compact, connected, oriented, CR manifold of hypersurface type in C^n is extended to a "strip" complex variety Y in C^n. The extension is obtained as a union of discs attached to M at points of local minimality (Trepreau-Tumanov); this extension "propagates" to points where minimality fails by a generalized Hanges-Treves theorem on account of the fact that these points are connected to the formers by a CR orbit. From a variant of the Hans Lewy theorem, we know that analytic sets extend across pseudoconcave boundaries, and from the Sperling-Rothstein theorem the different "patch extensions" glue up to a complex variety W sheeted overe C^n. Altogether, we obtain the extension of M to a complex variety W, that is, the Harvey-Lawson theorem. If M is pseudoconvex, then W encounters Y before touching its boundary M; in particular, W is smooth in a neighborhood of M. (If, in addition, M is contained in a pseudoconvex boundary to which it is not complex tangential), then in fact W belongs to C^n.) Once one realizes M as a boundary taken in a smooth way, one gets the answer to a conjecture by Kohn: the range of the di-bar tangential to M is closed.