The Hartpgs-type extension theorem for meromorphic mappings into q-complete complex spaces

Abstract

We prove in this note a result on extension of meromorphic mappings, which can be considered as a direct generalisation of the Hartogs extension theorem for holomorphic functions. Namely: THEOREM. Every meromorphic mapping $f:H_n^q(r)\to Y$, where $Y$ is a $q$ - -complete complex space, extends to a meromorphic mapping from $\Delta^{n+q}$ to $Y$. Here $H_n^q(r):=\Delta^n\times (\Delta^q\setminus \bar\Delta_r^q)\cup \Delta_r^n\times \Delta^q$ is a "q-concave" Hartogs figure in $C^{n+q}$. Remark that in the case $q=1$, i.e. when $Y$ is Stein, the statement of the Theorem is exactly the Theorem of Hartogs.