Extension and convergence theorems for families of normal maps in several complex variables

Abstract

Let H ( X , Y ) ( C ( X , Y ) ) \mathcal {H}(X,Y) ( \mathcal {C}(X,Y) ) represent the family of holomorphic (continuous) maps from a complex (topological) space X X to a complex (topological) space Y Y , and let Y + = Y ∪ { ∞ } Y^{+} = Y \cup \{\infty \} be the Alexandroff one–point compactification of Y Y if Y Y is not compact, Y + = Y Y^{+}=Y if Y Y is compact. We say that F ⊂ H ( X , Y ) \mathcal {F} \subset \mathcal {H}(X,Y) is uniformly normal if { f ∘ φ : f ∈ F \{f \circ \varphi : f \in \mathcal {F} , φ ∈ H ( M , X ) } \varphi \in \mathcal {H}(M,X)\} is relatively compact in C ( M , Y + ) \mathcal {C}(M,Y^{+}) (with the compact–open topology) for each complex manifold M M . We show that normal maps as defined and studied by authors in various settings are, as singleton sets, uniformly normal families, and prove extension and convergence theorems for uniformly normal families. These theorems include (1) extension theorems of big Picard type for such families – defined on complex manifolds having divisors with normal crossings – which encompass results of Järvi, Kiernan, Kobayashi, and Kwack as special cases, and (2) generalizations to such families of an extension–convergence theorem due to Noguchi.