I was reading this notes, and there is some things that is unclear to me about the defition of the degree of a line bundle, page 16. First of all, here is the construction:
Let $L$ be a complex line bundle on Riemann surface $C$. Consider a general section $\sigma : C \rightarrow L$. We can produce such a section by giving it locally and then gluing it together using a partition of unity. Locally, the line bundle $L$ is trivial, so it looks like $\mathbb{C}\times \Delta\rightarrow \Delta$, where $\Delta$ is the open unit disk. In this local picture, $\sigma$ is just a map $\Delta \rightarrow \mathbb{C}$.
By perturbing $\sigma$ we can insist that it is transverse to the zero section.Then locally, the inverse image of $0 \in \mathbb{C}$ under the map $\sigma: \Delta \rightarrow \mathbb{C}$ is a finite number of points. Each point $p \in \mathbb{C}$ where $\sigma$ intersects the zero section is called a zero of $\sigma$. Around each such point p the section $\sigma$ is a map $\sigma: \Delta \rightarrow \mathbb{C}$ where $p=0 \in \Delta$ and $\sigma(0)=0$. The differential $d\sigma:T_0\Delta \rightarrow T_0\mathbb{C}$ is nonsingular two-by-two matrix. Notice that there was an ambiguity since the map $\sigma: \Delta \rightarrow \mathbb{C}$ is defined up to post-multiplication by $\mathbb{C}^{*}$. Fortunately, multiplying by a complex number does not change the sing of $\mathrm{det}\mbox{ }d\sigma$.Definition: The degree of $L$ is $\mathrm{deg}(L)=\sum_{p}\mathrm{sgn}(p) \in \mathbb{Z}$, where the sum is over all points where a transverse section $\sigma$ is zero.
My questions:
Q1) In the construction is used that $\sigma$ is of a specific type, satisfying that in the trivialization, as a function of $\Delta \rightarrow \mathbb{C}$, $\sigma$ is differentiable. But, in the definition he doesn't mention it. Why is it not important?
Q2) Why the definition doesn't depend on the section I take?
Here's another question that is not about the definition, but an application of it:
Q3) How can I show using this definition that the degree of the tangent bundle on a Riemann surface of genus $g$ is $2-2g$.
Best Answer
For oriented real vector bundles, the Euler class is dual to the intersection of a generic section and the 0-section. For this you can learn it from Bott-Tu, chapter 2 or section 5 from the following notes of Hutchings: https://math.berkeley.edu/~hutching/teach/215b-2011/cup.pdf
This should answer your Q2 and Q3.
For your Q1, it's just a matter of convention. I think when you talk about complex line bundles, it is a priori a smooth vector bundle. Thus all the sections you talk about would be smooth sections.