Editing Chern class (section)

=== Complex projective space ===
There is an exact sequence of sheaves/bundles:<ref>The sequence is sometimes called the [[Euler sequence]].</ref>
<math display="block">0 \to \mathcal{O}_{\mathbb{CP}^n} \to \mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus (n+1)} \to T\mathbb{CP}^n \to 0</math>
where <math>\mathcal{O}_{\mathbb{CP}^n} </math> is the structure sheaf (i.e., the trivial line bundle), <math>\mathcal{O}_{\mathbb{CP}^n}(1)</math> is [[Serre's twisting sheaf]] (i.e., the [[hyperplane bundle]]) and the last nonzero term is the [[tangent sheaf]]/bundle.

There are two ways to get the above sequence:

{{Ordered list
|<ref>{{harvnb|Hartshorne|loc=Ch. II. Theorem 8.13.}}</ref> Let <math>z_0, \ldots , z_n</math> be the coordinates of <math>\Complex^{n+1},</math> let <math>\pi\colon \Complex^{n+1} \setminus \{0\} \to \Complex\mathbb{P}^n</math> be the canonical projection, and let <math>U = \mathbb{CP}^n \setminus \{ z_0 = 0\}</math>. Then we have:
<math display="block">\pi^* d(z_i / z_0) = {z_0 dz_i - z_i d z_0 \over z_0^2}, \, i \ge 1.</math>
In other words, the [[cotangent sheaf]] <math>\Omega_{\Complex\mathbb{P}^n}|_U</math>, which is a free <math>\mathcal{O}_U</math>-module with basis <math>d(z_i / z_0)</math>, fits into the exact sequence
<math display="block"> 0 \to \Omega_{\Complex\mathbb{P}^n}|_U \overset{dz_i \mapsto e_i}\to \oplus_1^{n+1} \mathcal{O}(-1)|_U \overset{e_i \mapsto z_i}\to \mathcal{O}_U \to 0, \, i \ge 0,</math>
where <math>e_i</math> are the basis of the middle term. The same sequence is clearly then exact on the whole projective space and the dual of it is the aforementioned sequence.
|Let ''L'' be a line in <math>\Complex^{n+1}</math> that passes through the origin. It is an exercise in [[elementary geometry]] to see that the complex tangent space to <math>\Complex\mathbb{P}^n</math> at the point ''L'' is naturally the set of linear maps from ''L'' to its complement. Thus, the tangent bundle <math>T\Complex\mathbb{P}^n</math> can be identified with the [[hom bundle]]
<math display="block">\operatorname{Hom}(\mathcal{O}(-1), \eta)</math>
where η is the vector bundle such that <math>\mathcal{O}(-1) \oplus \eta = \mathcal{O}^{\oplus (n+1)}</math>. It follows:
<math display="block">T\Complex \mathbb{P}^n \oplus \mathcal{O} = \operatorname{Hom}(\mathcal{O}(-1), \eta) \oplus \operatorname{Hom}(\mathcal{O}(-1), \mathcal{O}(-1)) = \mathcal{O}(1)^{\oplus(n+1)}.</math>
}}

By the additivity of total Chern class <math>c = 1 + c_1 + c_2 + \cdots</math> (i.e., the Whitney sum formula),
<math display="block">c(\Complex\mathbb{P}^n) \overset{\mathrm{def}}= c(T\mathbb{CP}^n) = c(\mathcal{O}_{\Complex\mathbb{P}^n}(1))^{n+1} = (1+a)^{n+1},</math>
where ''a'' is the canonical generator of the cohomology group <math>H^2(\Complex\mathbb{P}^n, \Z )</math>; i.e., the negative of the first Chern class of the tautological line bundle <math>\mathcal{O}_{\Complex\mathbb{P}^n}(-1)</math> (note: <math>c_1(E^*) = -c_1(E)</math> when <math>E^*</math> is the dual of ''E''.)

In particular, for any <math>k\ge  0</math>,
<math display="block">c_k(\Complex\mathbb{P}^n) = \binom{n+1}{k} a^k.</math>