On complex manifolds exhausted by biholomorphic images of generalized complex ellipsoids $\mathbb{E}$ (n;n₁, ... , n_s;p₁, ... , p_s)

Abstract

We assume that a complex manifold M can be exhausted by biholomorphic images of a complex manifold N, that is, for any compact set K in M, there exists a biholomorphic mapping f_K from N into M such that K⊂f_K(N). Then, how can we describe M using the data of N? We can see many articles related closely to this problem. The purpose of this note is to study this problem in the case when N is a generalized complex ellipsoid $\mathbb{E}\left( {n;n_1 ,..,n_s \,;\,p_1 ,..,p_s } \right)\, = \,\{ \left( {z_1 ,..,z_s } \right) \in \mathbb{C}^{n_1 } \times .. \times \mathbb{C}^{n_s } = \mathbb{C}^n \,;\,|\,z_1 \,|^{2p_1 } \, + .. + \,|\,z_s \,|^{2p_s } \, < 1\}$, where 0 < p₁,..,p_s ∈ ℝ and 0 < n₁, n_s ∈ Z with n₁ +..+n _s = n.

This work was done while the author was visiting University of California, Berkeley. He would like to thank the Mathematics Department for its warm hospitality.