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I've often read that the first chern class can be seen as "the number of zeroes a section must have".

How precise can this statement be made?

I'm only interested in Line bundles. I actually know how a generic section looks like on my manifold. From this I can even tell what the "lowest" number of zeros is.

Is it safe to say that

minimum number of zeros $> 0$ $\to$ $c_1 \neq 0$?

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1 Answer 1

up vote 3 down vote accepted

If a line bundle has a non-zero section, it is a trivial line bundle and that tells you what the $c_1$ is. Conversely, if every section has a zero, then the bundle is not trivial and since complex line bundles are actually classified by their first Chern class, then $c_1$ cannot be zero.

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Thanks! Wanted to make sure :) –  Mike Aug 21 '12 at 18:42
"complex line bundles are actually classified by their first Chern class" Is this true for bundles over any base space? –  Earthliŋ Jun 12 at 0:59
It is true if by cohomology you mean Cech cohomology and construct Chern classes there. For other cohomologies when they do not coincide with Cech's, I dunno. –  Mariano Suárez-Alvarez Jun 12 at 1:05
Thanks. Any hint as to why this is true? –  Earthliŋ Jun 12 at 1:12
You can describe line bundles using 1-cocycles in the Cech cohomology of sufficiently fine coverings with values in the multiplicative group of the complex numbers. This gives you a rule to attach a class in $H^1(X,\mathbb C^\times)$ to each complex line bundle. Then use the long exact sequence for cech cohomology on the short exact sequence of coefficient groups $0\to\mathbb Z\to\mathbb C\to\mathbb C^\times\to0$ to get a class in $H^2(X,\mathbb Z)$. –  Mariano Suárez-Alvarez Jun 12 at 1:56

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