Gauge fixing for logarithmic connections over curves and the Riemann-Hilbert problem. (English) Zbl 0978.14037
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.
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.
Reviewer: Lucian Bădescu (Bucureşti)
MSC:
14H55 | Riemann surfaces; Weierstrass points; gap sequences |
30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |
14H60 | Vector bundles on curves and their moduli |