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

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

2010-03-25

arxiv.org/abs/1003.4835v1

Mathematics

Complex Variables

18 pages

Scientific paper

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.

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