A vanishing theorem and asymptotic regularity of powers of ideal sheaves

Let I be an ideal sheaf on Pn. In the first part of this paper, we bound the asymptotic regularity of powers of I as , where e is a constant and s is the s-invariant of I. We also give the same upper bound for the asymptotic regularity of symbolic powers of I under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of I when it defines a local complete intersection subvariety with log canonical singularities.