Background.
In section 27.5 of the Mirror Symmetry book, the following claim is made:
The normal bundle of any embedding $\mathbb{P}^1 \subseteq X$ (where $X$ is a Calabi–Yau threefold) is a rank $2$, degree $-2$ bundle.
The given reasoning is roughly:
- $X$ is Calabi–Yau, so $K_X$ is trivial.
- $\int_{\mathbb{P}^1} c_1(T\mathbb{P}^1) = 2$.
From this, the claim follows.
My thought process.
The first point follows from the definition of a Calabi-Yau threefold, and I understand how to obtain the second point from the Euler sequence:
$$ 0 \to \mathcal O \to \mathcal O(1)^{\oplus 2} \to T\mathbb{P}^1 \to 0, $$
which gives $c_1(T\mathbb{P}^1) = 2H$ and then $\int_{\mathbb{P}^1} c_1(T\mathbb{P}^1) = 2$.
- There is the tangent–normal exact sequence
$$ 0 \to TC \to i^\ast TX \to N_{C\mid X} \to 0, $$
which comes from the map $i:\mathbb{P}^1\hookrightarrow X$, and $C$ is the image of $\mathbb P^1$.
- $i^\ast TX$ is a rank $3$ bundle, since it is the restriction of $TX$, and $\dim X = 3$. $TC$ is a line bundle, so $N_{C\mid X}$ has rank $2$.
- I know the degree of the bundle might be related to an integral so I expect the second point above to be relevant, but I’m blanking on the relation.
At this point I get stuck: how do we deduce $\deg N_{C\mid X} = -2$ and where am I using that $K_X \cong \mathcal O_X$? Does this have to do with Serre duality?
Also: do you have any recommendations for accessible references (something “hand-holding” like) that could help me work through the Mirror Symmetry book more effectively? My background: a couple of courses in AG and one in Riemann surfaces, no Hartshorne though :(
