Polystable saturated reflexive parabolic sheaves admit admissible Hermitian-Einstein metrics compatible with the parabolic structure, and semistable ones admit approximate such metrics; a Bogomolov-Gieseker inequality for nef and big classes follows.
Fibr\'es paraboliques et champ des racines
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abstract
Following ideas of Nori, Biswas, ..., we show that given an integer r>0, a noetherian scheme X, and an effective Cartier divisor D on it, the parabolic vector bundles on (X,D) with weights multiples of 1/r (in the sense of Maruyama-Yokogawa) are equivalent to ordinary vector bundles on an orbifold, the stack of r-th roots associated to (X,D) (a twisted scheme in the sense of Abramovich-Vistoli). We use this fact to get some information on the finite parabolic bundles on the (pointed) projective line.
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Kobayashi-Hitchin Correspondence for Saturated Reflexive Parabolic Sheaves on K\"ahler manifolds
Polystable saturated reflexive parabolic sheaves admit admissible Hermitian-Einstein metrics compatible with the parabolic structure, and semistable ones admit approximate such metrics; a Bogomolov-Gieseker inequality for nef and big classes follows.