  In the present article, we establish an equality condition in the restriction
formula on jumping numbers by giving a sharp lower bound of the dimension of
the support of a related coherent sheaf. As applications, we obtain equality
conditions in the restriction formula on complex singularity exponents by
giving the dimension, the regularity and the transversality of the support, and
we also obtain some sharp equality conditions in the fundamental subadditivity
property on complex singularity exponents. We also obtain two sharp relations
on jumping numbers.
