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Let E be a Complex vector bundle of rank r on M. Given $s_1, \cdots , s_r$ generic global sections, i can characterize the i-th Chern class as follows:

$ C_i(E)= \eta_{V_i}$, and $V_i$ is the locus in M where $s_1, \cdots, s_i$ are not linearly indipendent

So if $s_1, \cdots s_i$ are everywhere linearly indipendent $C_i(E)=0$, in this way Chern classes measure the distance of a fiber bundle from being trivial.

Now i'm applying this characterization to the Whitney product formula on E,F vector bundles of rank $\geq 2$ on M:

$C(E \bigoplus F)=C(E)C(F)$, so, for example

$C_2(E\bigoplus F)=C_2(E)+C_2(F)+C_1(E)C_1(F)$

but this appeared strange to me, because $C_2(E\bigoplus F)=0$ if there are 2 not collinear sections on $E \bigoplus F$, while the right side of the equality is zero if there are 2 not collinear sections on E, two on F, and one section of E or F always non zero.

Then i thought that this could be because $C_2(E\bigoplus F)$ expresses a more "global" distance of $E\bigoplus F$ from being trivial, that is $E \bigoplus F$ is trivial if and only if both E and F are trivial and so $C_2(E\bigoplus F)$ is zero if both E and F have two always not collinear global sections.

Am i right or i misunderstood?

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    The true statement I allude to above should be something like "If $V$ has enough holomorphic sections, and $c_1(V)=0$, then $V$ is trivial." I don't know a precise statement of this.2012-01-31

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