  We extend Nadel's results on some conditions for the multiplier ideal sheaves
to satisfy which are described in terms of an obstruction defined by the first
author. Applying our extension we can determine the multiplier ideal sheaves on
toric del Pezzo surfaces which do not admit K\"ahler-Einstein metrics. We also
show that one can define multiplier ideal sheaves for K\"ahler-Ricci solitons
and extend the result of Nadel using the holomorphic invariant defined by Tian
and Zhu.
