On Tautness Modulo an Analytic Subset of Complex Spaces
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On Tautness Modulo an Analytic Subset of Complex Spaces Pham Viet Duc1 · Mai Anh Duc2 · Pham Nguyen Thu Trang2
Received: 21 December 2016 / Revised: 8 February 2017 / Accepted: 28 February 2017 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017
Abstract The main goal of this article is to give necessary and sufficient conditions on the tautness modulo an analytic subset of complex spaces. Keywords Hyperbolicity modulo an analytic subset · Tautness modulo an analytic subset Mathematics Subject Classification (2010) Primary 32M05 · Secondary 32H02 · 32H15 · 32H50
1 Introduction The notions of hyperbolicity and tautness modulo an analytic subset of complex spaces are due to S. Kobayashi (see [1, p. 68]). Much attention has been given to these notions, and the results on related problems can be applied to many areas of mathematics, in particular to the extensions of holomorphic mappings. For details, see [1] and [2]. The main goal of this article is to give necessary and sufficient conditions on the tautness modulo an analytic subset of complex spaces.
Mai Anh Duc
[email protected] Pham Viet Duc [email protected] Pham Nguyen Thu Trang [email protected] 1
Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen Str., Thai Nguyen, Vietnam
2
Department of Mathematics, Hanoi National University of Education, 136 XuanThuy Str., Hanoi, Vietnam
P. V. Duc et al.
First of all, we recall the definitions of the tautness modulo an analytic subset of complex spaces . |a − b| Let be the open unit disc in the complex plane and ρ(a, b) := tan h−1 be the |1 − ab| Poincare distance on . Definition 1.1 (see [2, p. 240]) Let X be a complex space and S be an analytic subset in X. We say that X is taut modulo S if it is normal modulo S, i.e., for every sequence {fn } in H ol(, X) one of the following holds: i. There exists a subsequence of {fn } which converges uniformly to f ∈ H ol(, X) in H ol(, X); ii. The sequence {fn } is compactly divergent modulo S in H ol(, X), i.e., for each compact set K ⊂ and each compact set L ⊂ X \ S, there exists an integer N such that fn (K) ∩ L = ∅ for all n ≥ N . If S = ∅, then X is said to be taut. By the definition, it is easy to see that if S ⊂ S ⊂ X and X is taut modulo S, then X is taut modulo S . In particular, if X is taut, then X is taut modulo S for any analytic subset S of X.
2 Criteria on the Tautness Modulo an Analytic Subset of Complex Spaces First of all, we show the inheritance of the tautness modulo an analytic subset of complex spaces under certain assumptions. Theorem 2.1 Let X be a complex space and S be an analytic subset in X. Assume that X= Xi , where {Xi } are irreducible components of X. Then, X is taut modulo S if and i∈I
only if Xi is taut modulo Si := Xi ∩ S for all i ∈ I . Proof Assume that {Xi }i∈I are irreducible components of X. Then, {Xi }i∈I is a locally H ol(, Xi ). finite family and H ol(, X) = i∈I
(⇒) Assume that X is taut modulo
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