Several Complex Variables and the Geometry of Real Hypersurfaces

A Remark on Two Notions of Order of Contact

“…In [6], D'Angelo made precise the relationship between the 1-type ∆ 1 of the germ of a smooth hypersurface in C n and the corresponding notion of 1-type T 1 for holomorphic ideals in O n . We consider here a situation in which this relationship is particularly simple.…”

“…These measurements arise when defining the order of contact of q-dimensional complex analytic varieties with a real hypersurface in C n ( [5], [3]). The work in [6] shows how to reduce such questions about the local geometry of a real hypersurface to questions about ideals in the ring of germs of holomorphic functions at a point. We therefore carry out our work in the holomorphic setting, and in the last part of the paper we indicate the consequences in the hypersurface case.…”

Finite type points on subsets of Cn

“…This implies that We should note that after local biholomorphic change of coordinates, although the ideal I(U, k, p 0 ) changes, the inequality (1.6) still holds. Indeed, Corollary 3 on page 65 in [5] implies that D(I) is invariant under a local biholomorphic change of coordinates. Also, τ (I) is invariant under a local biholomorphic change of coordinates, (see the remarks after Definition 8 on page 72 in [5]).…”

“…Indeed, Corollary 3 on page 65 in [5] implies that D(I) is invariant under a local biholomorphic change of coordinates. Also, τ (I) is invariant under a local biholomorphic change of coordinates, (see the remarks after Definition 8 on page 72 in [5]). Then it follows from a similar chain of inequalities as above that ∆(M, p) ≤ 2(∆(M, p 0 )) n−d for all p ∈ V := V 0 ∩ V 1 .…”

“…If a type property fails for hypersurfaces, it also fails in higher codimension. For example, in [5], D'Angelo gave an example of the hypersurface…”