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It's known that the moduli stack ${\cal M}$ of semistable sheaves on a given polarized projective variety, with a fixed Hilbert polynomial, is compact (meaning that ${\cal M}$ has an atlas of finite type and that any family of such objects over a smooth punctured curve extends over the puncture). Current proofs I can found all go through the GIT setting. Q. Is there a proof in existence (at least in some special cases) that uses more directly the stability condition (cf. Bridgeland's work) without having to resorting to GIT? Best thanks --CHL

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