A cohomological criterion for semistable parabolic vector bundles on a curve

Biswas, Indranil (2007) A cohomological criterion for semistable parabolic vector bundles on a curve Comptes Rendus Mathematique, 345 (6). pp. 325-328. ISSN 1631-073X

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S16310...

Related URL: http://dx.doi.org/10.1016/j.crma.2007.07.004

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 Ex of this type is parabolic semistable if and only if there is a parabolic vector bundle Fx, also of this type, such that the underlying vector bundle (Ex⊗ Fx⊗ Vx)0 for the parabolic tensor product Ex⊗ Fx⊗ Vx)0 is cohomologically trivial, which means that Hi(X,(Ex⊗ Fx⊗ Vx)0) = 0 for all i. Given any parabolic semistable vector bundle Ex, the existence of such Fx 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.

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