D’Angelo finite type

Let $M\subset{\mathbb{C}}^{n}$ be a smooth submanifold of real codimension 1. Let $p\in M$ and let $r_{p}$ denote the generator of the principal ideal of germs at $p$ of smooth functions vanishing on $M$ . Define the number

\Delta_{1}(M,p)=\sup_{z}\frac{v(z^{*}r_{p})}{v(z)},

where $z$ ranges over all parametrized holomorphic curves $z\colon{\mathbb{D}}\to{\mathbb{C}}^{n}$ (where ${\mathbb{D}}$ is the unit disc) such that $z(0)=0$ , $v$ is the order of vanishing at the origin, and $z^{*}r_{p}$ is the composition of $r_{p}$ and $z$ . The order of vanishing $v(z)$ is $k$ if $k$ is the smallest integer such that the $k$ th derivative of $z$ is nonzero at the origin and all derivatives of smaller order are zero at the origin. Infinity is allowed for $v(z)$ if all derivatives vanish.

We say $M$ is of (or finite 1-type) at $p\in M$ in the sense of D’Angelo if

\Delta_{1}(M,p)<\infty.

If $M$ is real analytic, then $M$ is finite type at $p$ if and only if there does not exist any germ of a complex analytic subvariety at $p\in M$ , that is contained in $M$ . If $M$ is only smooth, then it is possible that $M$ is not finite type, but does not contain a germ of a holomorphic curve. However, if $M$ is not finite type, then there exists a holomorphic curve which “touches” $M$ to infinite order.

The Diederich-Fornaess theorem can be then restated to say that every compact real analytic subvariety of ${\mathbb{C}}^{n}$ is of D’Angelo finite type at every point.

References

1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
2 D’Angelo, John P. , CRC Press, 1993.

Title	D’Angelo finite type
Canonical name	DAngeloFiniteType
Date of creation	2013-03-22 17:39:57
Last modified on	2013-03-22 17:39:57
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Definition
Classification	msc 32V35