On the contact between complex manifolds and real hypersurfaces in 𝐶³

Abstract

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