Holomorphic vector bundles on the Riemann sphere and the 21st Hilbert problem (Q1356281)

Let \(D=\{a_1,\dots,a_n\}\) be a finite subset of the Riemann sphere \(P^1({\mathbb{C}})\) and let \[ \chi:\pi_1(P^1({\mathbb{C}})\setminus D)\to GL(p,\mathbb{C}) \] be a representation. The so-called Riemann-Hilbert problem (RHP) consists in finding a Fuchsian differential system \[ df=\sum_{i=1}^n \left(B_i\frac{dz}{z-a_i}\right)f \] having \(\chi\) as its monodromy group, where \(B_i\) are constant \(p\times p\) matrices. The representation \(\chi\) determines a vector bundle \(F\) on \(P^1({\mathbb{C}}) \setminus D\) with connection \(\nabla\). The author defines a new invariant (maximal weight) of \(\chi\) using all extensions of \(F\) onto \(P^1({\mathbb{C}})\) such that \(\nabla\) has at most simple poles on \(D\). Then if follows the theorem: If the maximal weight of the representation \(\chi\) is finite, then the RHP has a positive solution.