Let \(\Sigma\) be a compact Riemann surface of genus \(g\), \(X=\{x_1,\dots,x_n\}\) a finite subset of \(\Sigma\). A logarithmic connection over \((\Sigma,X)\) consists of a vector bundle \(E\) of rank \(r\) on \(\Sigma\) endowed with a holomorphic connection \(\nabla:\Omega^0(E)\to\Omega^0(E)\otimes\Omega^1 _{\Sigma}(\log X)\). One can naturally associate to such a logarithmic connection the monodromy map \(\chi:\pi_1(\Sigma\setminus X)\to \text{GL}(r,\mathbb C)\). A logarithmic connection is called quasi-fuchsian if the splitting type of the vector bundle \(E\) satisfies certain natural inequalities.
Then the main question coming from Hilbert’s twenty first problem is the following: For which \((\Sigma,X)\) and maps \(\chi:\pi_1(\Sigma\setminus X)\to \text{GL}(r,\mathbb C)\) there exists a quasi-fuchsian logarithmic connection realising \((X,\chi)\)? Sometimes this problem is also called Riemann-Hilbert problem.
The authors prove several results in this direction (especially when \(\Sigma\) is the projective line) by a gauge-theoretic approach using ideas of P. Deligne [“Equations différentielles à points singuliers réguliers”, Lect. Notes Math. 163 (1970; Zbl 0244.14004)] and C. T. Simpson [J. Am. Math. Soc. 3, 713-770 (1990; Zbl 0713.58012)]. The authors recover and generalize in particular some earlier results of A. A. Bolibruch and V. P. Kostov.