Local cohomology multiplicities in in terms of \'etale cohomolgy

We give a description of certain local cohomology invariants (introduced by Lyubeznik) for a class of mildly singular varieties in positive characteristic. The novelty of our approuch lies in using the new Riemann-Hilbert type correspondenc of Emerton and Kisin to express these invariants in terms of 'etale cohomology with ZZ/pZZ coefficients. Thus we are able to obtain results in positive characteristic whereas earlier approaches were restricted (due to the use of de Rham theory) to the analytic case.