Editing Chern class (section)

=== Normal sequence ===
Computing the characteristic classes for projective space forms the basis for many characteristic class computations since for any smooth projective subvariety <math>X \subset \mathbb{P}^n</math> there is the short exact sequence
<math display="block">0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^n}|_X \to \mathcal{N}_{X/\mathbb{P}^n} \to 0</math>

==== Quintic threefold ====
For example, consider a nonsingular [[quintic threefold]] in <math>\mathbb{P}^4</math>.  Its normal bundle is given by <math>\mathcal{O}_X(5)</math> and we have the short exact sequence
<math display="block">0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^4}|_X \to \mathcal{O}_X(5) \to 0</math>

Let <math>h</math> denote the hyperplane class in <math>A^\bullet(X)</math>. Then the Whitney sum formula gives us that
<math display="block">c(\mathcal{T}_X)c(\mathcal{O}_X(5)) = (1+h)^5 = 1 + 5h + 10h^2 + 10h^3 </math>

Since the Chow ring of a hypersurface is difficult to compute, we will consider this sequence as a sequence of coherent sheaves in <math>\mathbb{P}^4</math>. This gives us that
<math display="block">\begin{align}
c(\mathcal{T}_X) &= \frac{1 + 5h + 10h^2 + 10h^3}{1 + 5h} \\
&= \left(1 + 5h + 10h^2 + 10h^3\right)\left(1 - 5h + 25h^2 - 125h^3\right) \\
&= 1 + 10h^2 - 40h^3
\end{align}</math>

Using the Gauss-Bonnet theorem we can integrate the class <math>c_3(\mathcal{T}_X)</math> to compute the Euler characteristic. Traditionally this is called the [[Euler class]]. This is
<math display="block">\int_{[X]} c_3(\mathcal{T}_X) = \int_{[X]} -40h^3 = -200</math>
since the class of <math>h^3</math> can be represented by five points (by [[Bézout's theorem]]). The Euler characteristic can then be used to compute the Betti numbers for the cohomology of <math>X</math> by using the definition of the Euler characteristic and using the Lefschetz hyperplane theorem.
<!-- Cite https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf page 181-182 -->

==== Degree d hypersurfaces ====
If <math>X \subset \mathbb{P}^3</math> is a degree <math>d</math> smooth hypersurface, we have the short exact sequence <math display="block">0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^3}|_X \to \mathcal{O}_X(d) \to 0</math> giving the relation <math display="block">c(\mathcal{T}_X) = \frac{c(\mathcal{T}_{\mathbb{P}^3|_X})}{c(\mathcal{O}_X(d))}</math> we can then calculate this as
<math display="block">\begin{align}
c(\mathcal{T}_X) &= \frac{(1+[H])^4}{(1 + d[H])} \\
&= (1 + 4[H] + 6[H]^2)(1-d[H]+d^2[H]^2) \\
&= 1 + (4-d)[H] + (6-4d+d^2)[H]^2
\end{align}</math>
Giving the total chern class. In particular, we can find <math>X</math> is a spin 4-manifold if <math>4-d </math> is even, so every smooth hypersurface of degree <math>2k</math> is a [[spin manifold]].