On direct images of twisted pluricanonical sheaves on normal varieties

We study the depth properties of certain direct image sheaves on normal varieties. Let f : Y -> X be a proper morphism of relative dimension d from a smooth variety onto a normal variety such that the preimage E of the singular locus of X is a divisor. We show that for any integer m > 0, the higher direct image R-d f*omega Y-circle times m (aE) modulo the torsion subsheaf is S-2, provided that a is sufficiently large. In case f is birational, we give criteria on a for the direct image f*omega Y (aE) to coincide with.X. We also introduce an index measuring the singularities of normal varieties.