Line bundles on the moduli space of parabolic connections over a compact Riemann surface

Abstract

Let X be a compact Riemann surface of genus g≥3 and S a finite subset of X. Let ξ be fixed a holomorphic line bundle over X of degree d. Let Mpc(r,d,α) (respectively, Mpc(r,α,ξ)) denote the moduli space of parabolic connections of rank r, degree d and full flag rational generic weight system α, (respectively, with the fixed determinant ξ) singular over the parabolic points S⊂X. Let Mpc′(r,d,α) (respectively, Mpc′(r,α,ξ)) be the Zariski dense open subset of Mpc(r,d,α) (respectively, Mpc(r,α,ξ)) parametrizing all parabolic connections such that the underlying parabolic bundle is stable. We show that there is a natural compactification of the moduli spaces Mpc′(r,d,α), and Mpc′(r,α,ξ) by smooth divisors. We describe the numerically effectiveness of these divisors at infinity. We determine the Picard group of the moduli spaces Mpc(r,d,α), and Mpc(r,α,ξ). Let C(L) denote the space of holomorphic connections on an ample line bundle L over the moduli space M(r,d,α) of parabolic bundles. We show that C(L) does not admit any non-constant algebraic function.