image of pseudoconvex domain is not pseudoconvex?

Question

Explicitly, how can we find functions and domains satisfy the following:
(i) Both $U$ and $V$ are domains of $\mathbb{C}^n$.
(ii) $U$ is pseudoconvex.
(iii) there exists a surjective holomorphic map between $U$ and $V$.
(iv) $V$ is not pseudoconvex.

Answer 1

Let $U=\mathbb{C}^n$ and $V=\mathbb{C}^n\setminus\{0\},\ n\geq 2,$ we will finish it if we can contruct a surjective holomorphic map between $U$ and $V$ by Hartogs' Theorem.
Define $$f\colon U\rightarrow V,\ (z_1,\cdots,z_n)\mapsto (z_1,\cdots,z_{n-1},z_1z_n+e^{z_n}).$$ By Picard's Little Theorem, we know for any $a\in \mathbb{C}\setminus\{0\}$, the map $$\mathbb{C}\rightarrow \mathbb{C}, z\mapsto e^z +az$$ is surjective, hence $f$ is surjective.