The local polynomial hull near a degenerate CR singularity -- Bishop discs revisited

TL;DR

This paper investigates the local polynomial convexity of smooth real surfaces in complex space near non-parabolic CR singularities, establishing a complete criterion based on a Maslov-type index for the existence of Bishop discs.

Contribution

It provides a complete characterization of Bishop discs around non-parabolic CR singularities using a Maslov-type index, extending previous partial results.

Findings

Presence of Bishop discs is determined by a Maslov-type index.

The result generalizes all known facts for order-two, non-parabolic CR singularities.

Refines ideas from Bishop beyond potential theory.

Abstract

Let S be a smooth real surface in C^2 and let p\in S be a point at which the tangent plane is a complex line. How does one determine whether or not S is locally polynomially convex at such a p --- i.e. at a CR singularity ? Even when the order of contact of T_p(S) with S at p equals 2, no clean characterisation exists; difficulties are posed by parabolic points. Hence, we study non-parabolic CR singularities. We show that the presence or absence of Bishop discs around certain non-parabolic CR singularities is completely determined by a Maslov-type index. This result subsumes all known facts about Bishop discs around order-two, non-parabolic CR singularities. Sufficient conditions for Bishop discs have earlier been investigated at CR singularities having high order of contact with T_p(S). These results relied upon a subharmonicity condition, which fails in many simple cases. Hence, we look beyond potential theory and refine certain ideas going back to Bishop.