weakly holomorphic

Let V be a local complex analytic variety. A function f:UV (where U is open in V) is said to be weakly holomorphic through U if there exists a nowhere dense complex analytic subvariety WV and W contains the singular pointsMathworldPlanetmathPlanetmath of V and VWU, and such that f is holomorphic on VW and f is locally bounded on V.

It is not hard to show that we can then just take W to be the set of singular points of V and have U=VW as we can extend f to all the nonsingular points of V.

Usually we denote by 𝒪w(V) the ring of weakly holomorphic functions through V. Since any neighbourhood of a point p in V is a local analytic subvariety, we can define germs of weakly holomorphic functions at p in the obvious way. We usually denote by 𝒪pw(V) the ring of germs at p of weakly holomorphic functions.


  • 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
Title weakly holomorphic
Canonical name WeaklyHolomorphic
Date of creation 2013-03-22 17:41:46
Last modified on 2013-03-22 17:41:46
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 4
Author jirka (4157)
Entry type Definition
Classification msc 32C15
Classification msc 32C20
Synonym w-holomoprhic
Related topic NormalComplexAnalyticVariety