Definition 1Let X be a variety and let x be a point of X. The multiplicity of x on X, denoted mult , means mult , .For a normal variety X, a prime divisor E over X means that a prime divisor E appears on a resolution f :Y → X. Let E be a prime divisor over X and appear on a log resolution f :Y →X of X. The log discrepancy of X with respect to E isFor a closed subset Z of X, the minimal log discrepancy mld (X ) over Z is the infimum of (X ) for all prime divisor E over X with center in Z. Conjecture 1Let X be an -dimensional locally a complete intersection variety. Then mult ≦2

We define the ω-multiplier ideals on a normal variety. The main goal of this paper is to introduce an ω-multiplier ideal and prove its properties. We give characterizations of two-dimensional rational singularities by means of ω-multiplier ideals and cores of ideals.

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We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.

This paper answers in the affirmative a question raised by Huneke and Watanabe concerning an upper bound on the multiplicity of a normal Cohen-Macaulay Du Bois singularity.