On the contact between complex manifolds and real hypersurfaces in $\textbf {C}^{3}$

Abstract:

Let $m$ be a real ${\mathcal {C}^\infty }$ hypersurface of an open subset of ${{\mathbf {C}}^3}$ and let $p \in M$. Let ${a^1}(M,p)$ denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of $p$ in ${{\mathbf {C}}^3}$ with $M$ at $p$. Let ${c^1}(M,p)$ denote the $\sup \{ m \in {\mathbf {Z}}|$ for all tangential holomorphic vector fields $L$ with $L(p) \ne 0$ then ${L^{{i_0}}}{\bar L^{{j_0}}} \ldots {L^{{i_n}}}{\bar L^{{j_n}}}({\mathfrak {L}_M}(L))(p) = 0\}$ where ${i_0}, \ldots ,{i_n};{j_0}, \ldots ,{j_n}$ are positive integers such that $\sum \nolimits _{t = 0}^n {{i_t} + {j_t} = m - 3}$ and ${\mathfrak {L}_M}(L)$ denotes the Levi form of $M$ evaluated on the vector field $L$. Theorem. If $M$ is pseudoconvex near $p \in M$ then ${a^1}(M,p) = {c^1}(M,p)$.