A nullset for normal functions in several variables

Abstract

Suppose that Ω \Omega is a domain in C n , E ⊂ Ω {C^n},E \subset \Omega is closed in Ω \Omega , and f : Ω ∖ E → C ∗ f:\Omega \backslash E \to {C^ * } is a meromorphic function. We show that if f f is normal and E E is an analytic subvariety or, more generally, of locally finite ( 2 n − 2 ) (2n - 2) -dimensional Hausdorff measure in Ω \Omega satisfying a certain geometric condition, then f f can be extended to a meromorphic function (= holomorphic mapping) f ∗ : Ω → C ∗ {f^ * }:\Omega \to {C^ * } . In the case of a subvariety sufficient, but not necessary, for the geometric condition is that the singularities of E E are normal crossings. As a digression, we give a new proof for the following result, due to Parreau in the case n = 1 n = 1 : if f f is in the Nevanlinna class and E E is polar (in R 2 n {R^{2n}} ), then f f has a meromorphic extension f ∗ {f^ * } to Ω \Omega .