Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor

We give a criterion for the projectivisation of reflexive sheaf on klt space to be induced by projective representation fundamental group smooth locus. This is then applied characterisation finite quotients spaces and Abelian varieties byQ\mathbb {Q}-Chern class (in)equalities suitable stability condition. condition formulated in terms naturally defined extension tangent structure sheaf. further examine cases which this satisfied, comparing it K-semistability related notions.