On Stein neighborhood basis of real surfaces

Abstract

Abstract. In this paper, we show that a compact real surface embedded in a complex surface has a regular Stein neighborhood basis, provided that there are only finitely many complex points on the surface, and that they are all flat and hyperbolic. An application to unions of totally real planes in C 2 is then given.

Polynomially convex embeddings of even-dimensional compact manifolds

“…We note that the construction of p.s.h. defining functions with additional properties is of independent interest in the literature ( [27]), and is related to the existence of regular Stein neighbourhood bases. and is strictly p.s.h.…”

“…Although, elliptic and hyperbolic points are both stable under small C 2 -deformations, a surface is locally polynomially convex only near its hyperbolic complex points. In [27], Slapar proves a (possibly stronger) result for flat hyperbolic points ( [13]), i.e., when local holomorphic coordinates can be chosen so that Im o(|z| 2 ) ≡ 0, i.e., M is locally contained in C × R. It is shown that, near a flat hyperbolic p, M is the zero set of a nonnegative function that is strictly p.s.h. in its domain except at p.…”

“…Each slice S α ∩ {Z ′ ∈ C 2k : (w 1 , ..., w 2k−2 ) = (s 1 , ..., s 2k−2 )}, where (s 1 , ..., s 2k−2 ) ∈ R 2k−2 , is a totally real surface with an isolated flat hyperbolic complex point at the origin in C 2 z,w . These have been studied by Slapar in [27]. A slight modification of his construction yields the following key ingredient of our proof.…”

Stein structures and holomorphic mappings

“…The point p is elliptic if λ < 1/2 and hyperbolic if λ > 1/2; the degenerate case λ = 1/2 does not arise for a generic M. At an elliptic point M has a nontrivial local envelope of holomorphy consisting of a family of small analytic discs (the so called Bishop discs, [3]). At a hyperbolic point M is locally holomorphically convex [24] and it admits a basis of tubular Stein neighborhoods [53].…”

“…By [16,Theorem 2.2] this new M admits a basis of smoothly bounded Stein neighborhoods diffeomorphic to M × R 2 , given as sublevel sets of a smooth plurisubharmonic function τ ≥ 0 which vanishes precisely on M and has no critical points in a deleted neighborhood of M. (Locally at a special hyperbolic point w = z 2 +z 2 we can take τ (z, w) = |w − z 2 −z| 2 . It was shown in [53] how to find strongly pseudoconvex tubular neighborhoods of any surface with only hyperbolic complex points. )…”