# D’Angelo finite type

Let $M\subset {\u2102}^{n}$ 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 $z:\mathbb{D}\to {\u2102}^{n}$ (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

$$ |

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 ${\u2102}^{n}$ 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 |