An extension theorem for separately meromorphic functions with pluripolar singularities

Details An extension theorem for separately meromorphic functions with pluripolar singularities An extension theorem for separately meromorphic functions with pluripolar singularities

2002-09-17

arxiv.org/abs/math/0209207v2

Kyushu J. Math. 57 (2003), 291-302.

Mathematics

Complex Variables

11 pages

Scientific paper

Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset X$ be relatively closed. For any $j\in\{1,...,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1})\times(A_{j+1}\times...\times A_N)$ such that the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb C^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,...,\Sigma_N$ are pluripolar. Put ${multline*} X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times...\times A_{j-1})\times D_j \times(A_{j+1}\times...\times A_N): (z',z'')\notin\Sigma_j\}$. Then there exists a relatively closed pluripolar subset $\widetilde M\subset\widetilde X$ of the `envelope of holomorphy' $\widetilde X$ of $X$ such that: $\bullet$ $\widetilde M\cap X'\subset M$, $\bullet$ every function $f$ separately meromorphic on $X\setminus M$ extends to a (uniquely determined) function $\widetilde f$ meromorphic on $\widetilde X\setminus\widetilde M$, $\bullet$ if $f$ is separately holomorphic on $X\setminus M$, then $\widetilde f$ is holomorphic on $\widetilde X\setminus\widetilde M$, and $\bullet$ $\widetilde M$ is singular with respect to the family of all functions $\widetilde f$. \noindent In the case where N=2, $M=\varnothing$, the above result may be strengthened.

Affiliated with

Also associated with

No associations

Rating

An extension theorem for separately meromorphic functions with pluripolar singularities does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with An extension theorem for separately meromorphic functions with pluripolar singularities, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An extension theorem for separately meromorphic functions with pluripolar singularities will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-589228