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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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    $\begingroup$ 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. $\endgroup$ May 28, 2013 at 20:19
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    $\begingroup$ 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. $\endgroup$
    – Sam Lisi
    May 29, 2013 at 0:19

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From purely algebraic arguments:

  1. If $E \to X$ is a complex vector bundle of complex dimension $n$, then the top Chern class $c_n(E) \in H^{2n}(X; \mathbb R)$ is equal to the Euler class $e(E)$.
  2. Every $n$-dimensional complex manifold $X$ (such as $\mathbb {CP}^n$) is orientable. In algebraic terms, orientability can be seen as the top homology group $H_{2n}(X; \mathbb R)$ having rank one, and we can choose an orientation, i.e. a choice of generator $[X] \in H_{2n}(X; \mathbb R)$.

Now let $\mathbb {CP}^n = X$ and $x$ be the Euler class of the tautological line bundle over $\mathbb {CP}^n$. Then there is a nice argument on p22 Example 2.9.3 of these lecture notes on characteristic classes (from the algebraic topology point of view). Essentially, it boils down to showing that

$$ \langle e(T \mathbb {CP}^n), [\mathbb {CP}^n] \rangle =(n+1)(-1)^n \langle x^n, [\mathbb {CP}^n] \rangle$$

where $\langle,\rangle : H^*(\mathbb {CP}^n, \mathbb R) \otimes_{\mathbb R} H_*(\mathbb {CP}^n, \mathbb R) \to \mathbb R$ is the canonical non-degenerate pairing between homology and cohomology and $x^n$ is the cup product of $x$ with itself $n$ times.

We know from standard theory that for an orientable manifold $M$, the pairing $\langle e(TM), [M] \rangle = \chi(M)$ -- the Euler characteristic. The homology groups of $\mathbb {CP}^n$ have rank one in every even grading that is $\leq 2n$ and zero otherwise, so $\chi(M)= n+1$. Hence, $x\neq 0$.

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  • $\begingroup$ could you please provide that link again, the current one is not in a valid url. $\endgroup$
    – Eden Zane
    Nov 12, 2023 at 9:06

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