Let $X$ be a complex manifold (or algebraic variety) and let $p:L\to X$ be a line bundle. Given $s_0,\ldots,s_n\in H^0(X,L)\setminus \{0\}$ non-zero global sections we can define the meromorphic (resp. rational) map
$$\varphi_L:X\dashrightarrow \mathbb{P}^n,\;x\mapsto [s_0(x),\ldots,s_n(x)] $$
(This can be also defined in a more intrinsic way without coordinates, but here I am looking for a reality check…). Let us assume that the $s_j$ have no common zeroes (i.e. $L$ is globally generated). Then, it is well-known that $\varphi_L^*\mathcal{O}_{\mathbb{P}^n}(1)\cong L$.
A nice proof of this fact, using linear equivalence of divisors, can be found on Huybrechts' book "Complex Geometry", Proposition 2.3.26 on page 85.
Here, however I would like to ask if it is possible to check this just by definition (using transition functions). More precisely, we know that the line bundle $\mathcal{O}_{\mathbb{P}^n}(1)$ can be trivialized on the standard open subsets $U_i = \{x\in \mathbb{P}^n\;|\;x_i\neq 0 \}$ and the transition functions from $U_j$ to $U_i$ is given by $\psi_{ij}(x)=x_j/x_i$ (c.f. Huybrechts' book, Proposition 2.2.6 on page 68). On the other hand, we have by definition that
$$\varphi_L^\ast \mathcal{O}_{\mathbb{P}^n}(1) = \{(x,\ell)\in X \times \mathcal{O}_{\mathbb{P}^n}(1)\;|\;\varphi_L(x)=[s_0(x),\ldots,s_n(x)]=p(\ell)\} $$
is the pullback line bundle.
From this, we can check that $\varphi_L^\ast \mathcal{O}_{\mathbb{P}^n}(1)$ can be trivialized on the open set $X_i:=\varphi_L^{-1}(U_i)=\{x\in X\;|\;s_i(x)\neq 0\}$ so we are finished if we are able to prove that the transition funcions $h_{ij}$ from $X_j$ to $X_i$ coincide with the transition functions of our original line bundle $L$ (and for this, maybe we would have to restrict ourselves to a finer open cover, right?).
Now, if for simplicity we assume that the original line bundle $L$ can be trivialized on the $X_i$ and that the transition functions from $X_j$ to $X_i$ are given by $g_{ij}$, then we have that $s_i = g_{ij}s_j$ (c.f. Huybrechts's book, Definition 2.3.23 on page 84). On the other hand, the transition functions $h_{ij}$ of the pullback $\varphi_L^\ast \mathcal{O}_{\mathbb{P}^n}(1)$ are given by
$$h_{ij}=\varphi_L^\ast (\psi_{ij}) = \varphi_L^\ast\left(\frac{x_j}{x_i} \right)= \frac{x_j \circ \varphi_L}{ x_j \circ \varphi_L} = \frac{s_j}{s_i}=\frac{s_j}{g_{ij}s_j}=\frac{1}{g_{ij}}=g_{ji}. $$
Therefore, for some reason I obtain the transition functions of the dual line bundle $L^\vee$. What am I missing ?
Thanks a lot in advance!
Best Answer
There are two different notions at play here, both with the same notation $s_i$. When you related them to the transition functions, these are holomorphic $\Bbb C$-valued functions representing a fixed section with respect to the trivializations on $U_i$. On the other hand, when you defined the mapping to projective space, the $s_i$ represent actual holomorphic sections of the line bundle $L$.
(The inverse relation you stumbled upon is the same inverse that occurs when you compare transformation of basis [frame] to transformation of coordinates in linear algebra.)