On singular projective structures on Riemann surfaces

Abstract

1 Let be a compact Riemann surface of genus g with function held J?(X) and let L=D’+AD+B (1.1) be a second order linear differential operator on X: D is a non-trivial derivation of A(X)/@ and A, B E A(X). We assume the singularities of L on to be all regular and denote by S a finite subset of containing all of them. Let X’ denote the punctured Riemann surface X S and let x0 E X’ be an ordinary point of L. Analytic continuation along closed paths of a ratio z of independent solutions of L at x0 yields a (class of) representa- tion(s) of the fundamental group x1(X’, x0) in the projective linear group PGL(2, C), which we call the projective monodromy representation pL of L. We may regard pL as a (conjugacy class of) group homomorphism(s) pr: x1(X’, x0) + PGL(2, a=) (1.2) or, via the natural identification, valid for any group G [S, Lemma 271 Hom(n,(X’, x,), G)/GrH’(X’, G), (1.3) as a cohomology class