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Let m be a real G°° hypersurface of an open subset of C3 and let p e M. Let a'(M,p) denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of p in C3 with M at p. Let cl(M,p) denote the sup{m e Zj for all tangential holomorphic vector fields L with L(p) ^ 0 then L'°/> . . . Z>Z>(ßM(L)X/>) = 0} where /„.i„; jo, ■ ■ ■ J„ are positive integers such that 2"_0 i, + j, = m — 3 and £.M(L) denotes the Levi form of M evaluated on the vector field L. Theorem. // M is pseudoconvex near p G M then a'(M,p) = cl(M,p). Introduction. The notion of type of a point on a real hypersurface in C2 was introduced by J. J. Kohn [12]. In [1] this notion was generalized to C. The type of a point p on a real hypersurface M is an integer > 2 or + oo. It may be characterized geometrically as the maximal order of contact of a nonsingular complex hypersurface with the real hypersurface M at p. It may be characterized algebraically as the minimal length of commutator of the tangential holomorphic vector fields and their conjugates needed to obtain the "missing direction". For domains D in C2 the type of a boundary point is an important invariant. For D pseudoconvex with smooth boundary bD, Kohn showed [12] that if p E. bD is of type m there is a local subelliptic estimate in the 9-Neumann problem of order l/m. P. Greiner showed that this is the best possible estimate [11] (see also [16], [18]). Recent progress [8], [15] has established subelliptic estimates on pseudoconvex domains in C with smooth real-analytic boundaries. The type of a boundary point gives the best possible subelliptic estimate for (p, n — 1) forms. However it is clear that in C (n > 2) other invariants will be needed. The work of Diederich-Fornaess and Kohn suggests that the order of contact of (possibly singular) subvarieties with bD at a point p £\ bD is an important invariant. (The best possible order of subelliptic estimates in the 9-Neumann problem or even sufficient conditions in the case of Q°° boundaries are currently open problems [5].) In this paper we consider a real pseudoconvex hypersurface M in C3 and the order of contact with one-dimensional complex submanifolds with M at p £\ M. Received by the editors November 30, 1979 and, in revised form, February 15, 1980. 1980 Mathematics Subject Classification. Primary 32F20, 32F25.

On the contact between complex manifolds and real hypersurfaces in 𝐶³