Holomorphic line bundles on $\mathbb{P}^1$ from gluing data

Question

Let $g$ be an $n \times n$ matrix of functions $g_{ij}(z)$ in $\mathbb{C}(z)$. Suppose that the $g_{ij}(z)$ have no poles on the annulus $1-\epsilon < |z| < 1+\epsilon$ and that $\det g(z)$ is nonzero on this annulus. Then I can define a holomorphic vector bundle on $\mathbb{P}^1$ by taking trivial rank $n$ bundles on $\{ |z|<1+\epsilon \}$ and $\{ |z| > 1-\epsilon \}$ and gluing them by $g$. This bundle must be isomorphic to $\bigoplus \mathcal{O}(d_i)$ for some unique $d_1 \geq d_2 \geq \cdots \geq d_n$.

How do I read off the $d_i$ from the $g_{ij}(t)$?

For comparison, I know a simple criterion when the annulus is replaced with all of $\mathbb{C} \setminus \{ 0 \}$: $d_1+d_2+\cdots + d_k$ is the minimum, over all $k \times k$ minors of $g$, of the order of vanishing of that minor at $0$.

Answer 1

I am just posting my comment above as an answer. Let $U$ and $V$ be subsets of $\mathbb{CP}^1$, or any other projective algebraic curve $C$, whose complement sets are finite, and such that $\{U,V\}$ is an open cover. Let $g=[g_{i,j}]$ be an $n\times n$ matrix of rational functions on $C$ that are regular on $U\cap V$ and such that $\text{det}(g)$ is nonzero on $U\cap V$. There is an associated locally free $\mathcal{O}_C$-module $\mathcal{E}$ together with isomorphisms, $$\phi:\mathcal{O}_U^{\oplus n} \to \mathcal{E}|_U, \ \ \psi:\mathcal{O}_V^{\oplus n} \to \mathcal{E}|_V,$$ such that the transition function $\psi\circ \phi^{-1}$ on $U\cap V$ is given by the matrix $[g_{i,j}]$. For the reduced divisor $\Delta$ that is the complement of $U\cap V$, for all integers $M\gg 0$, the induced maps, $$ \mathcal{O}_C(-M\underline{\Delta})^{\oplus n}|_U \hookrightarrow \mathcal{O}_U^{\oplus n} \xrightarrow{\phi} \mathcal{E}|_U, \ \ \mathcal{O}_C(-M\underline{\Delta})^{\oplus n}|_V \hookrightarrow \mathcal{O}_V^{\oplus n} \xrightarrow{\psi} \mathcal{E}|_V,$$ extend to morphisms of $\mathcal{O}_C$_modules, $$\phi_M:\mathcal{O}_C(-M\underline{\Delta})^{\oplus n} \to \mathcal{E}, \ \ \psi_M:\mathcal{O}_C(-M\underline{\Delta})^{\oplus n} \to \mathcal{E}.$$

For me, it is more direct to think about this when I form the "twisted transpose" of these maps, $$phi^\dagger_M:\mathcal{E}^\vee(-M\underline{\Delta}) \to \mathcal{O}_C^{\oplus n}, \ \ \psi^\dagger_M:\mathcal{E}^\vee(-M\underline{\Delta}) \to \mathcal{O}_C^{\oplus n}.$$ Now the problem is to simply understand, for each point $p\in \Delta$, the stalk of $\mathcal{E}(-M\underline{\Delta})$ as a $\mathcal{O}_{C,p}$-submodule of $\mathcal{O}_{C,p}^{\oplus n}$ containing the submodule $(\mathfrak{m}_{C,p}^M)^{\oplus n}$ via $\phi$, resp. via $\psi$. There is a local description of this involving only the integer $M$ and the image of $[g_{i,j}]$ in $\text{Mat}_{n\times n}(\mathcal{O}_{C,p})$.

For every finite subset $\Delta\subset C$, for every integer $M$, for the data at each point $p\in \Delta$ of a $\mathcal{O}_{C,p}$-submodule $E_p$ of $\mathcal{O}_{C,p}^{\oplus n}$ that contains $(\mathfrak{m}_{C,p}^M)^{\oplus n}$, there is a unique $\mathcal{O}_C$-submodule $\mathcal{E}^\vee(-M\underline{\Delta})$ of $\mathcal{O}_C^{\oplus n}$. The problem of understanding this is purely algebraic. As suggested by Sasha, for $C=\mathbb{CP}^1$, we can simply work out which global sections of $\mathcal{O}_C(d)^{\oplus n}$ are contained in the submodule $\mathcal{E}^\vee(-M\underline{\Delta})(d)$ for each integer $d=0,1,\dots,M\text{deg}(\Delta)$ in terms of the vanishing conditions at each point $p\in \Delta$ imposed by the local submodule constraints. This will uniquely determine the splitting type of $\mathcal{E}^\vee(-M\underline{\Delta})$. If you know more about $\Delta$ and the local submodule data, you can say more, e.g., if $\Delta=\{0,\infty\}$ and the submodules are $\mathbb{C}^\times$-invariant then the splitting type is straightforward. For fixed $M$ and $\text{deg}(\Delta)$, there is a parameter space of "local data", and there is a stratification of this parameter space into locally closed subsets on which the splitting type is constant. I doubt we can describe these strata "explicitly" once $M$ and $\text{deg}(\Delta)$ are large.

Historically, I believe that Birkhoff had some papers on this phrased just in terms of matrices whose entries are rational functions. I believe this is how Birkhoff proved what is often called "Grothendieck's splitting lemma" (only for $\textbf{GL}_n$ -- Grothendieck proved his theorem for all semisimple groups). This was also understood by Bertini and del Pezzo via their explicit classification of minimal varieties (excluding Veronese surfaces) as joins of rational curves.