Analytic discs under sympletic transforms

Abstract

By a holomorphic homogeneous symplectic transformation of T*X (for X = ℂ^N), we interchange the conormal bundle T ^*_M X to a higher codimensional submanifold M with the conormal bundle T ^*_M X to a hypersurface M of X. For an analytic disc A “attached” to M we are able to find a section A* ⊂T*X with π A* = A, attached to T ^*_M X, such that Ã:= πx(A*) is an analytic disc “attached” to M. By this procedure of “transferring” analytic discs, we get the higher codimensional version of our criteria of [5] on holomorphic extension of CR functions (with [5] being on its hand the main tool of the present proof). Thus, let W be a wedge of X with generic edge M and assume that there exists an analytic disc contained in M ∪ W, tangent to M at a boundary point z₀∈ ∂A, and not contained in M in any neighborhood of z₀. Then germs of holomorphic functions on W at z₀ extend to a full neighborhood of z₀.