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

Multiplier ideal sheaves, jumping numbers, and the restriction formula