On the monodromy map for the logarithmic differential systems

Abstract

We study the monodromy map for logarithmic g-differential systems over an oriented surface S 0 of genus g, with g being the Lie algebra of a complex reductive affine algebraic group G. These logarithmic g-differential systems are triples of the form (X, D, Φ), where (X, D) ∈ T g,d is an element of the Teichmüller space of complex structures on S 0 with d ≥ 1 ordered marked points D ⊂ S 0 = X and Φ is a logarithmic connection on the trivial holomorphic principal G-bundle X × G over X whose polar part is contained in the divisor D. We prove that the monodromy map from the space of logarithmic g-differential systems to the character variety of G-representations of the fundamental group of S 0 \ D is an immersion at the generic point, in the following two cases:(1) g ≥ 2, d ≥ 1, and dim C G ≥ d + 2;(2) g = 1 and dim C G ≥ d. The above monodromy map is nowhere an immersion in the following two cases:(1) g = 0 and d ≥ 4;(2) g ≥ 1 and dim C G < d+3g−3 g . This extends to the logarithmic case the main results in [CDHL], [BD] dealing with nonsingular holomorphic g-differential systems (which corresponds to the case of d = 0).