G. Giorgadze

Institute of Cybernetics, Georgian Academy of Sciences, Tbilisi, Georgia

FUCHSIAN SYSTEMS ON RIEMANN SURFACES

Dedicated to the memory of Andrey Bolibruch

The fundamental work of A.Bolibruch gave a new impetus to the study of Fuchsian systems [1]. In particular, A.Bolibruch invented some powerful tools for investigation of the famous Riemann-Hilbert problem and global properties of Fuchsian system in terms of associated holomorphic vector bundles with meromorphic connections. This technique later was successfully applied for solving the Riemann-Hilbert problem for compact Riemann surfaces of higher genus (see [2] and references therein). The results presented in the talk belong to the same direction.

Denote by Λ_{X}^{1}(log S) the sheaf of 1-forms holomorphic over X \ S, where X is a compact Riemann surface and S is a finite subset. We consider admissible pairs over (X, S) consisting of a holomorphic vector bundle E → X and holomorphic connection ∇ : Λ^{0}(E) → Λ^{0} ⊗ Λ_{X}^{1}(log S). For such pairs one can define the monodromy representation and splitting type [1]. A logarithmic connection (E, ▽) is called quasi-Fuchsian if there exists a splitting of E such that −2g(j − 1) ≤ k_{j−1} − k_{j} ≤ 2g, i = 1, …, n − 1. If k_{1} = 0, then logarithmic connection (E, ∇) is Fuchsian. The following theorem explicates some previously known results of Fuchsian systems with prescribed monodromy.

Theorem. 1) If the monodromy matrix ρ(γ_{j}) is diagonalisable for some j, then for representation ρ there exists a Fuchsian system whose monodromy representation coincides with ρ.

2) For any two dimensional representation ρ there exists a rank-two Fuchsian system whose monodromy representation coincides with ρ.

3) Let (E, ∇) be a stable pair. Then there exists a semi-stable pair (E′, ∇′) such that deg(E′) = 0, ∇′ has the same singular points as ∇, and the monodromy representations induced by the ∇ and ∇′ coincide.

We also present a similar result for holomorphic principal G-bundles with holomorphic connection and G-system on Riemann surfaces where G is a compact Lie group (see [3]). Such generalization of Fuchsian systems opens wide perspectives for studying certain differential equations of modern mathematical physics, for example, the two-dimensional Yang–Mills equations.

References

1. A. A. Bolibruch, The Riemann–Hibert problem. (Russian) Russian Math. Surveys 45(1990), No. 2, 1–47.

2. A. A. Bolibruch, Stable vector bundles with logarithmic connections and Riemann–Hilbert problem. (Russian) Dokl. Akad. Nauk 381(2001), No. 1, 10–13.

3. G. Giorgadze, Regular systems on Riemann surfaces. J. Math. Sci. 118 (2003), No. 5, 5347–5399.