On the semiampleness of the positive part of CKM Zariski decompositions

We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.
v4: exposition and organization of the paper improved. To appear in "Math. Proc. Camb. Phil.Soc."