I am reading this paper A logarithmic $\bar\partial$-equation on a compact Kähler manifold associated to a smooth divisor.
The setting is as follows: let $X$ be a compact complex manifold and let $D=\sum_{i=1}^{r} D_i$ be a simple normal crossing divisor on $X$, namely, $D_i$, $1\leq i\leq r$, are smooth hypersurfaces on $X$ and intersect with each other transversely. Therefore, For any $z \in X$, which $k$ of these $D_i$ pass, we may choose local holomorphic coordinates $\left\{z^1, \cdots, z^n\right\}$ in a small neighborhood $U$ of $z=(0, \cdots, 0)$ such that
$$
D \cap U=\left\{z^1 \cdots z^k=0\right\}
$$
is the union of coordinates hyperplanes.
Now, let $L$ be a holomorphic line bundle over $X$ satisfying
$$ L^N=\mathcal{O}_X\left(\sum_{i=1}^r a_i D_i\right)$$
for some $a_i \in \mathbb{Z}, 1 \leq i \leq r$.
Then, in page 6, the author argues as follows:
Let $\sigma$ be the canonical meromorphic section of $\mathcal{O}_X\left(\sum_{i=1}^r a_i D_i\right)$, and let $e$ be a local frame of $L$ satisfying
$$
e^N=\prod_{i=1}^r\left(z^i\right)^{-a_i} \sigma .\,\,\,\,\,\,\,\,\,\,(*)
$$
I am confused about $(*)$, recall that a canonical meromorphic section $\sigma$ (see for example this post canonical meromorphic section)is a global meromorphic function such that
$$\text{div}(\sigma)|_U+\sum_{i=1}^{r}a_iD_i|_U\geq 0$$
So, shouldn't $(*)$ be
$$
e^N=\prod_{i=1}^r\left(z^i\right)^{a_i} \sigma?
$$
What am I missing or is it just a mistake? Thanks for any possible suggestions!
Best Answer
The mistake I made is that I confused with the local frame and local coordinate.
Now assume that we have a collection of frames/basics $\left\{\tilde{e}_i\right\}$ of $L^N$ on $X=\left\{U_i\right\}$, then we should have $\tilde{e}_i=\phi_{i j}^{-1} \tilde{e}_j$. Thus, following the notations in my question $e^N:=\tilde{e}_i=\frac{\sigma}{\prod_{i=1}^r\left(z^i\right)^{a_i}}$, which is just $(*)$.