D’Angelo finite type

Let $M\subset {ℂ}^n$ 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

$${\mathrm{\Delta }}_1(M,p)=\underset{z}{sup}\frac{v(z^{*}{r}_p)}{v(z)},$$

where $z$ ranges over all parametrized holomorphic curves $z:𝔻\to {ℂ}^n$ (where$𝔻$ 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.

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

If $M$ is real analytic, then $M$ is finite type at $p$ if and only ifthere 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 analyticsubvariety of ${ℂ}^n$ is of D’Angelo finite type at every point.