Line bundle for each point

Question

I am traing to understand the following part from Hitchins lecture notes

I just don't see how the line bundle here is defined. What is meant by "patch together"?

Answer 1

Well, it's a bit late, but for the sake of completeness, here is an answer: $U_0$ and $U_1$ form an open covering of $M$.

We know, that from a "1-cocycle of transition functions $(g_{ij})_{i,j}$ subordinate to an open covering $(U_i)_i$", where $g_{ij}: U_i \cap U_j \to \mathrm{GL}(r,\mathbb{C})$ holomorphic, we can construct an holomorphic vector bundle of rank r of the form $$E := \frac{\bigsqcup_{i}(U_i \times k^r)}{\sim},$$ where $(x,v)\sim(y,w) :\Leftrightarrow x = y$ and $\exists i,j : x\in U_i \cap U_j$ such that $g_{ij} (x)v=w$.

In our case, as there are only two open sets in our covering, we only need one transition function $g_{01} := z : U_0 \cap U_1 \to \mathbb{C}-\{0\} \cong \mathrm{GL}(1,\mathbb{C})$, which forms automatically a 1-cocycle by $$(U_0 \cap U_1) \times \mathbb{C} \to (U_0 \cap U_1) \times \mathbb{C}, \\ (m, w) \mapsto (m, g_{01}(m)w).$$ The equivalence relation, see above, is the "patching together" of $U_0 \times \mathbb{C}$ and $U_1 \times \mathbb{C}$.