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 $M=∪H_a$   where $H_a =\{z∈C^n : a\cdot z=0\}$ . 

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

Thanks a lot!

Ron