Microlocal complex foliation of R-Lagrangian CR submanifolds

Abstract

Let $X$ be a complex manifold, $M$ a real analytic submanifold of $X^{R}$, $T^{*}X$ the cotangent bundle to $X$, $T_{M}^{*}X$ the conormal bundle to $M$ in $X$. Assume that $T_{M}^{*}X$ is regular and $CR$ in $T^{*}X$. We then show that $T_{M^{*}}X$ is locally defined as the zero-set of the real and/or imaginary part of holomorphic symplectic coordinates of $T^{*}X$. It is well known that the similar description of $M$ in local complex coordinates of $X$ is true only if $M$ is Levi flat. As an application we obtain a generalization of the celebrated edge of the wedge Theorem.