Remmert-Stein theorem

For a complex analytic subvariety V and pV a regular point, let dimpV denote the complex dimension of V near the point p.

Theorem (Remmert-Stein).

Let UCn be a domain ( and let S be a complex analytic subvariety of U of dimensionMathworldPlanetmath m<n. Let V be a complex analytic subvariety of U\S such that dimpV>m for all regular points pV. Then the closureMathworldPlanetmathPlanetmath of V in U is an analytic variety in U.

The condition that dimpV>m for all regularPlanetmathPlanetmathPlanetmathPlanetmath p is the same as saying that all the irreducible componentsPlanetmathPlanetmath of V are of dimension strictly greater than m. To show that the restrictionPlanetmathPlanetmath on the dimension of S is “sharp,” consider the following example where the dimension of V equals the dimension of S. Let (z,w)2 be our coordinates and let V be defined by w=e1/z in 2S, where S is defined by z=0. The closure of V in 2 cannot possibly be analytic. To see this look for example at W=V¯{w=1}. If V¯ is analytic then W ought to be a zero dimensional complex analytic set and thus a set of isolated points, but it has a limit pointPlanetmathPlanetmath (0,1) by Picard’s theoremMathworldPlanetmath.

Finally note that there are various generalizationsPlanetmathPlanetmath of this theorem where the set S need not be a varietyMathworldPlanetmathPlanetmath, as long as it is of small enough dimension. Alternatively, if V is of finite volume, we can weaken the restrictions on S even further.


  • 1 Klaus Fritzsche, Hans Grauert. , Springer-Verlag, New York, New York, 2002.
  • 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
Title Remmert-Stein theorem
Canonical name RemmertSteinTheorem
Date of creation 2013-03-22 15:04:55
Last modified on 2013-03-22 15:04:55
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 9
Author jirka (4157)
Entry type Theorem
Classification msc 32A60
Classification msc 32C25
Synonym Remmert-Stein extension theorem
Related topic ChowsTheorem