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Hi all:

I am working on Functional Analysis. I encounter a topology problem in my study of spectrum of certain operators, and it has bothered me for quite some 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

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  • $\begingroup$ I suspect that if $M$ is a discrete set of hyperplanes, $H^1(M^c)$ would be spanned by logarithmic $1$-forms along the components as you're hoping. However, I don't think it would be finite dimensional in general, for example, it isn't for $M= \mathbb{Z}\subset \mathbb{C}$. $\endgroup$
    – Donu Arapura
    Commented Sep 1, 2010 at 14:18
  • $\begingroup$ I should explain that my intuition above comes from looking at the homology $H_1(M^c,\mathbb{Z})$. This should be generated by loops around the components. The log $1$-forms time ($1/2\pi i$) would be the dual basis. $\endgroup$
    – Donu Arapura
    Commented Sep 1, 2010 at 14:25

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