Timeline for An example of a complex manifold without a finite open cover
Current License: CC BY-SA 2.5
5 events
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| Mar 21, 2011 at 12:26 | comment | added | Georges Elencwajg | Dear Vamsi, you are not being daft at all! On the contrary, it is all my fault: I thought that I could find a proof of my guess by somehow adapting the proofs that $\mathbb P^n $ cannot be covered by $n$ open affines and that the tautological bundle on $\mathbb P^n (\mathbb R)$ cannot be trivialized topologically by $n$ open subsets but I failed. I have edited my post to emphasize its status as guess. I hope one of the incredibly competent users here will help us settle the question. | |
| Mar 21, 2011 at 12:12 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
changed "think" to "guessing" and added "but cannot prove" for emphasis.
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| Mar 21, 2011 at 2:25 | comment | added | Vamsi | I am probably being very daft here, but why does the first Chern class being huge prevent the existence of a finite number of charts? (is it similar to the argument that goes into proving that for compact manifolds, the first Chern class is the Poincare dual of the fundamental class of the divisor? If there are finitely many charts, we are allowed to integrate using a partition of unity and so on...) | |
| Mar 20, 2011 at 22:14 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
Named manifold "X". Changed "normal bundle" to "$\mathcal O_X(D)"
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| Mar 20, 2011 at 21:56 | history | answered | Georges Elencwajg | CC BY-SA 2.5 |