We consider smooth deformations of the $CR$ structure of a smooth $2$-pseudoconcave compact $CR$ submanifold $\textsf{ M}$ of a reduced complex analytic variety $\textsf{ X}$ outside the intersection $D\,{ \cap}\,\textsf{ M}$ with the support $D$ of a Cartier divisor of a positive line bundle $\texttt{ F}_{ \textsf{ X}}.$ We show that nearby structures still admit projective $CR$ embeddings. Special results are obtained under the additional assumptions that $\textsf{ X}$ is a projective space or a Fano variety.
Categories: math.CV math.AG math.AP math.DG