New constructions relating Real and Complex Contact Structure

Abstract

We establish new connections between real and complex contact geometry via embeddings of 3-manifolds into $\C^3$. We introduce a new \emph{contact wedge} construction combining two transverse real contact structures to make a new \emph{complex contact} structure, subject to obstructions measured by the Nijenhuis tensor and Dolbeault cohomology. Dually, we form a \emph{hedge} construction which extracts real contact structures from complex ones. Applying these tools, we prove that $\C^3$ admits uncountably many complex contact structures.