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 ∗ of this type is parabolic semistable if and only if there is a parabolic vector bundle F ∗ , also of this type, such that the underlying vector bundle ( E ∗ ⊗ F ∗ ⊗ V ∗ ) 0 for the parabolic tensor product E ∗ ⊗ F ∗ ⊗ V ∗ is cohomologically trivial, which means that H i ( X , ( E ∗ ⊗ F ∗ ⊗ V ∗ ) 0 ) = 0 for all i. Given any parabolic semistable vector bundle E ∗ , the existence of such F ∗ 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. To cite this article: I. Biswas, C. R. Acad. Sci. Paris, Ser. I 345 (2007).