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

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.


32V40 Real submanifolds in complex manifolds
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