Unitary Monodromies of Rank Two Fuchsian Systems with (n + 1) Singularities

Abstract

We study the unitarity of monodromies of rank two Fuchsian systems of SL type with (n + 1) regular singularities on the Riemann sphere, namely, we give a sufficient and necessary condition for the monodromy group to be conjugate to a subgroup of a special unitary group SU(p,q). When n ≥ 3, the moduli space of irreducible monodromies can be realized as an affine algebraic set in C^m for some m ∈ N. In this paper, we give a characterization and construction of unitary monodromies in terms of this affine algebraic set. The signatures of unitary monodromies are also classified.