D’Angelo finite type

Let Mn be a smooth submanifold of real codimension 1. Let pM and let rp denote the generator of the principal ideal of germs at p of smooth functionsMathworldPlanetmath vanishing on M. Define the number


where z ranges over all parametrized holomorphic curves z:𝔻n (where 𝔻 is the unit disc) such that z(0)=0, v is the order of vanishing at the origin, and z*rp is the composition of rp and z. The order of vanishing v(z) is k if k is the smallest integer such that the kth 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 pM 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 pM, 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 n is of D’Angelo finite type at every point.


  • 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