We give a formula for the Donaldson--Futaki invariants of certain type of semi test configurations, which essentially generalizes the Ross--Thomas slope theory . The positivity (resp. non-negativity) of those ``a priori special'' Donaldson--Futaki invariants implies K-stability (resp. K-semistability). As an application, we prove the K-(semi)stability of certain polarized varieties with semi-log-canonical singularities, which generalizes some results of .
"A generalization of the Ross--Thomas slope theory." Osaka J. Math. 50 (1) 171 - 185, March 2013.