characteristic polynomial of vector bundle

Question

I am trying to understand the following exerpt from Hitchin - Segal- Ward's book :

A point in the cotangent bundle is thus (up to equivalence) a vector bundle E and a holomorphic section A of $\operatorname{End}E\otimes K$.

Locally, A is just an $m \times m$ matrix with values in the canonical bundle, and we can take its characteristic polynomial $\operatorname{det}(w - A) = w^m + a_1w^{m-1} + • • • + a_m$ where now instead of each coefficient $a_j$ being a polynomial, it is a global holomorphic section of the line bundle $K^i$.

1. Why is $A$ with values in $K$ and why is $a_i\in K^i$, should I read $a_i \in K^{\otimes i}?$

2. I saw quite few versions of characteristic polynomials (like $\operatorname{det}(\lambda I- A), \operatorname{det}(I_d -A)$) and now $\operatorname{det}(w-A)$. Is there a logic behind these different versions?

3. What is the canonical bundle? (I guess it is the line bundle $K$)?

Thank you for your help.

Answer 1

The canonical bundle is the line bundle (invertible sheaf) $K$ of differential forms of top degree. In this case it is the cotangent bundle of the Riemann Surface. The characteristic polynomial is the usual one only the notation os bad, $\det(wI−A)$ is what they mean. And yes, $K^i=K^{\otimes i}$.

To understand why $A$ is locally a matrix with values in $K$ we only need first to observe that $$(\operatorname{End} E) \otimes K = \operatorname{Hom}(E,E\otimes K)$$to write the morphism $A\colon E \rightarrow E\otimes K$ over a local trivialization. Indeed on a small open set $U$ we may have $E|_U \simeq U\times \mathbb{C}^m$ and $A$ is given by a $m\times m$ matrix whose entries are local sections of $K$ i.e. one-forms.