  In this article, we obtain two sharp equality conditions in the restriction
formula on complex singularity exponents: an equality between the codimension
of the zero variety of related multiplier ideal sheaves and the relative
codimension of the restriction of the variety on the submanifold (in the
restriction formula); an equivalence between the transversality (between the
variety and the submanifold) and the regularity of the restriction of the
variety. As applications, we present sharp equality conditions in the
fundamental subadditivity property on complex singularity exponents.
