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.
Calculating the Parabolic Chern Character of a Locally Abelian Parabolic Bundle
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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.