chern class of universal line bundle

chern class of universal line bundle [closed]

consider the $\mathbb{C}P^{n}$ and the universal line bundle $E \rightarrow \mathbb{C}P^{n}$, where $$E = \{(l,v)| l \in \mathbb{C}P^{n}, v \in \mathbb{C}^{n+1}, v \in l \},$$ show $\langle c_{1}(E), [\mathbb{C}P^{1}]\rangle = -1$.

edited Jul 31 2011 at 6:50

algori
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asked Jul 31 2011 at 5:47

niko
1

This is really an elementary exercise, not something that belongs on this site, in my opinion. – Robert Bryant Jul 31 2011 at 6:06

how do i choose a connection ? – niko Jul 31 2011 at 6:08

niko -- I second Robert Bryant's remark. There are many ways to do this, depending on what one is allowed to use. Let me sketch one of them: restrict the bundle to a complex line and show that a section of the dual bundle has one transversal zero. You may also want to take a look at Milnor, Stasheff, Characteristic classes, chapter on Chern classes. – algori Jul 31 2011 at 6:54

closed as off topic by algori, Ryan Budney, Neil Strickland, José Figueroa-O'Farrill, Tim Perutz Jul 31 2011 at 20:46

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