On equality of line type and variety type of real hypersurfaces in \(\mathbb{C}^ n\). (English) Zbl 0749.32009

Boas, Harold P.; Straube, Emil J.

J. Geom. Anal. 2, No. 2, 95-98 (1992).

Let \(M\) be a smooth real hypersurface in \(\mathbb{C}^ n\) near a point \(p\) in \(M\). The type of \(p\) is the maximal order of contact of complex varieties with \(M\) at \(p\). In general the computation of this important geometric invariant can be quite complicated. In this paper the authors present a class of \(M\) for which the type at \(p\) can be computed using only complex affine lines through \(p\).
The class consists of all \(M\) for which there is a neighborhood of \(p\) such that for every real tangent ray at \(p\), the height of \(M\), in this neighborhood, above the ray is a nondecreasing function of the distance from \(p\). This class includes the boundaries of convex domains but is more general. For example the boundary of a domain which is starshaped with respect to the boundary point \(p\) is also in this class.
The exposition is very clear and the argument quite elementary and direct. The only perhaps lesser known technical tool used from the literature is a result relating the absolute value of a polynomial to that of its leading coefficient.

Reviewer: G.A.Harris (Lubbock)

MSC:

32V40

Real submanifolds in complex manifolds

Keywords:

type of a point; order of contact; real hypersurface

References:

[1]	Catlin, D. Necessary conditions for subellipticity of the \(\bar \partial - Neumann\) problem. Ann. Math.117, 147–171 (1983). · Zbl 0552.32017
[2]	Catlin, D. Subelliptic estimates for the \(\bar \partial - Neumann\) problem on pseudoconvex domains. Ann. Math.126, 131–191 (1987). · Zbl 0627.32013
[3]	D’Angelo, J. P. Real hypersurfaces, orders of contact, and applications. Ann. Math.115, 615–637 (1982). · Zbl 0488.32008
[4]	D’Angelo, J. P. Finite-type conditions for real hypersurfaces in Cn. In: Complex Analysis, Lecture Notes in Mathematics 1268, pp. 83–102. Berlin: Springer 1987.
[5]	Goluzin, G. M. Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs 26. Providence, RI: American Mathematical Society 1969. · Zbl 0183.07502
[6]	McNeal, J. D. Convex domains of finite type. Preprint. · Zbl 0777.31007
[7]	Pólya, G., and Szegö, G. Problems and Theorems in Analysis II. Berlin: Springer 1976. · Zbl 0359.00003