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I'm studying characteristic classes from the Chern-Weil construction (via connection and curvature). I'm trying to compute some simple examples. Let $E$ be the tautological line bundle over projective space $P(\mathbf{C}^n)$. I want to show that the first Chern class of $E$ does not vanish. I suppose I could just introduce a connection on $E$ using local trivializations (is there a natural choice?), patch things together, compute the curvature and from this the first Chern class. However, that sounds a bit tedious. Are there more elegant ways to compute it?

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I would endorse computing some examples from scratch. If you're familiar with the mechanism of moving frames in differential geometry, this is a snap. You could do it with transition functions, yes. –  Ted Shifrin May 28 '13 at 20:19
    
After you have done the computation this way, I think it would be instructive to see how Milnor-Stasheff do it. I learned a lot from working through that section of their book. –  Sam Lisi May 29 '13 at 0:19

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