How to distinguish the tautological line bundle and the trivial line bundle on $P^n$?
How can I tell that these are not isomorphic as bundles?
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1Over which field? – PVAL-inactive Apr 18 '15 at 01:23
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And as what kind of bundles? Algebraic? Analytic? Topological? (It's a special feature of the case of $\mathbb{P}^n$ that these three classifications coincide, but this isn't true of varieties in general.) – Qiaochu Yuan Apr 18 '15 at 03:23
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Does the tautological line bundle have a nowhere-zero section $\sigma$? (Consider $\sigma([x]) = f(x)x$ for some continuous function $f\colon S^n\to\Bbb R$.) The same argument will work in the case of $\Bbb CP^n$ if you think a bit.
Ted Shifrin
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