# complement of an infinite union of hyperplanes

Hi all:

A topology problem has bothered me for quite a long time. Any idea or references is greatly appreciated.

Suppose $M$ is an infinite (possibly uncountable) union of complex hyperplanes in $C^n$. To be specific, we write $$M=\cup H_{a}$$ where $H_a=\{z\in C^n:\ a\cdot z=0\}$.

If $M$ is a finite union, then the de Rham cohomology (with complex coefficient) of $M^c$ is generated by the 1-forms $\frac{a\cdot dz}{a\cdot z}$. This is a well-known theorem. My question is whether there is a similar theorem for an infinite union of hyperplanes. We can assume the first de Rham cohomology $H^1(M^c,\ C)$ is finite dimensional. In particular, is $H^1(M^c,\ C)$ spanned by the 1-forms $\frac{a\cdot dz}{a\cdot z}$?

Thanks a lot!

Ron Yang

SUNY at Albany

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The set $M$ may not be closed (and so $M^c$ may not be open and thus may not have de Rham cohomology). – Robin Chapman Jul 12 '10 at 14:39
...and you probably mean a countable union of hyperplanes? – Mariano Suárez-Alvarez Jul 12 '10 at 14:56
By the way: why can we «assume that the first de Rham cohomology is finite dimensional»? – Mariano Suárez-Alvarez Jul 12 '10 at 15:04
There is a well-developed theory for the case of $\textit{affine}$ hyperplanes, but I cannot recall having seen anything interesting in the infinite linear case. – Victor Protsak Jul 13 '10 at 3:32