Monodromy, the Riemann-Hilbert Problem, and the Differential Galois Group
Let U be an open connected subset of the complex sphere P1=C∪{∞} and let Y′=AY be a differential equation on U, with A an n×n matrix with coefficients that are meromorphic functions on U. We assume that the equation is regular at every point p∈U. Thus, fo
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5.1 Monodromy of a Differential Equation Let U be an open connected subset of the complex sphere P1 = C ∪ {∞} and let Y = AY be a differential equation on U, with A an n × n matrix with coefficients that are meromorphic functions on U. We assume that the equation is regular at every point p ∈ U. Thus, for any point p ∈ U, the equation has n independent solutions y1 , . . . , yn consisting of vectors with coordinates in C({z − p}). It is known ([132], Chap. 9; [225], p. 5) that these solutions converge in a disk of radius ρ, where ρ is the distance from p to the complement of U. These solutions span an n-dimensional vector space denoted by V p . If we let F p be a matrix whose columns are the n independent solutions y1 , . . . , yn then F p is a fundamental matrix with entries in C({z− p}). One can normalize F p such that F p ( p) is the identity matrix. The question we are interested in is: Does there exist on all of U, a solution space for the equation having dimension n? The main tool for answering this question is analytical continuation which in turn relies on the notion of the fundamental group ([7], Chap. 8; [132], Chap. 9). These can be described as follows. Let q ∈ U and let λ be a path from p to q lying in U (one defines a path from p to q in U as a continuous map λ : [0, 1] → U with λ(0) = p and λ(1) = q). For each each point λ(t) on this path, there is an open set Oλ(t) ⊂ U and fundamental solution matrix Fλ(t) whose entries converge in Oλ(t) . By compactness of [0, 1], we can cover the path with a finite number of these open sets, {Oλ(ti ) }, t0 = 0 < t1 < · · · < tm = 1. The maps induced by sending the columns of Fλ(i) to the columns of Fλ(i+1) induce C-linear bijections Vλ(ti ) → Vλ(ti+1 ) . The resulting C-linear bijection V p → Vq can be seen to depend only on the homotopy class of λ (we note that two paths λ0 and λ1 in U from p to q are homotopic if there exists a continuous H : [0, 1] × [0, 1] → U such that H(t, 0) = λ0 (t), H(t, 1) = λ1 (t) and H(0, s) = p, H(1, s) = q). The C-linear bijection V p → Vq is called the analytic continuation along λ. For the special case that λ(0) = λ(1) = p we find an isomorphism that is denoted by M(λ) : V p → V p . The collection of all closed paths, starting and ending in p,
M. van der Put. et al., Galois Theory of Linear Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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5. Monodromy and Riemann-Hilbert
divided out by homotopy, is called the fundamental group and denoted by π1 (U, p). The group structure on π1 (U, p) is given by “composing” paths. The resulting group homomorphism M : π1 (U, p) → GL(V p ) is called the monodromy map. The image of M in GL(V p ) is called the monodromy group. The open connected set U is called simply connected if π1 (U, p) = {1}. If U is simply connected then one sees that analytical continuation yields n independent solutions of the differential equation on U. Any open disk, C and also P1 are simply connected. The fundamental group of U := {z ∈ C| 0 < |z| < a} (for a ∈ (0, ∞]) is generated by the circle around
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