Local polynomial convexity of the union of two totally real surfaces at their intersection

TL;DR

This paper investigates conditions under which the union of two totally real surfaces in complex space is locally polynomially convex at their intersection, addressing known cases and exploring new geometric obstructions.

Contribution

It provides a new approach to the problem, identifies obstructions to polynomial convexity, and discusses conditions affecting local polynomial convexity at intersections.

Findings

When tangent planes intersect trivially, local polynomial convexity holds.

Obstructions exist when the intersection of tangent spaces has dimension one.

Geometric conditions influence the local polynomial convexity of surface unions.

Abstract

We consider the following question: Let $S_1$ and $S_2$ be two smooth, totally-real surfaces in $\mathbb{C}^2$ that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is $S_1 \cup S_2$ locally polynomially convex at the origin? If $T_0S_1 \cap T_0S_2=\{0\}$, then it is a folk result that the answer is yes. We discuss an obstruction to the presumed proof, and provide a different approach. When dimension of $T_0S_1 \cap T_0S_2$ over the field of real numbers is 1, we present a geometric condition under which no consistent answer to the above question exists. We then discuss conditions under which we can expect local polynomial convexity.