Locally holomorphic sets and the Levi form

Louis Roberts Hunt

Vol. 42 (1972), No. 3, 681–688

DOI: 10.2140/pjm.1972.42.681

Abstract

Suppose we have a real k-dimensional 𝒞² manifold M embedded in Cⁿ. If M has a nondegenerate complex tangent bundle of positive rank at some point p ∈ M, then the vanishing or nonvanishing of the Levi form on M near p determines whether or not M is locally holomorphic at p. We show that if M is locally holomorphic at p, then the Levi form vanishes near p, the converse being a known result. In addition we prove a C − R extendibility theorem for a certain case when M is 𝒞^∞ and has a nonzero Levi form at p ∈ M.

Mathematical Subject Classification

Primary: 32F15

Milestones

Received: 7 May 1971

Revised: 29 October 1971

Published: 1 September 1972