LetX be a compact Riemann surface andM_s^p(X) the moduli space of stable parabolic vector bundles with fixed rank, degree, rational weights and multiplicities. There is a natural Kähler metric onM_s^p(X). We obtain a natural metrized holomorphic line bundle onM_s^p(X) whose Chern form equalsmr times the Kähler form, wherem is the common denominator of the weights andr the rank.

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,ϕ), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, andϕ: E₂→ E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.