What's an example of a set with trivial holomorphic first sheaf cohomology? I was thinking of subsets of C, and trying to think what would satisfy this.(adsbygoogle = window.adsbygoogle || []).push({});

For example, suppose you covered C by [tex]U_1=re^{i\theta}:r < 2[/tex] and [tex]U_2=re^{i\theta}:r> 1/2.[/tex] Then if [tex]f(z)=1/z[/tex], [tex]\oint_{|r|=1} f(z)\neq 0,[/tex] and he says the contour integral of a function in [tex]U_1\cap U_2[/tex] should be 0. So I don't think subdividing C this way is a good limiting open cover.

But what would be a good limiting open cover, other than just C by itself? One should be able to refine an arbitrary open cover of C into one which has trivial holomorphic first sheaf cohomology, since C by itself does.

Supposing you covered C by [tex]U_1=z: Re(z) > \pi/2[/tex] and [tex]U_2=z: Re(z) < \pi.[/tex] I don't see how that open cover would have trivial first sheaf cohomology either since [tex]\displaystyle e^{1/ sin(z)}[/tex] would be analytic in [tex]U_1\cap U_2[/tex] and I don't think [tex]e^{1/ sin(z)}[/tex] could be expressed as the sum of a function that's analytic in [tex]U_1[/tex] and a function that's analytic in [tex]U_2.[/tex] So would such an open cover need more refinement? Into what?

What's a good book on it?

Laura

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# Trivial holomorphic first sheaf cohomology

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