A cohomological criterion for semistable parabolic vector bundles on a curve

Abstract

Let X be an irreducible smooth complex projective curve and S ⊂ X a finite subset. Fix a positive integer N. We consider all the parabolic vector bundles over X whose parabolic points are contained in S and all the parabolic weights are integral multiples on 1/N. We construct a parabolic vector bundle V, of this type, satisfying the following condition: a parabolic vector bundle E_x of this type is parabolic semistable if and only if there is a parabolic vector bundle F_x, also of this type, such that the underlying vector bundle (E_x⊗ F_x⊗ V_x)₀ for the parabolic tensor product E_x⊗ F_x⊗ V_x)₀ is cohomologically trivial, which means that Hⁱ(X,(E_x⊗ F_x⊗ V_x)₀) = 0 for all i. Given any parabolic semistable vector bundle E_x, the existence of such F_x is proved using a criterion of Faltings which says that a vector bundle E over X is semistable if and only if there is another vector bundle F such that E⊗F is cohomologically trivial.