On direct images of twisted pluricanonical sheaves on normal varieties

We study the depth properties of certain direct image sheaves on normal varieties. Let $f:Y\to X$ 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 $R^d{f}_{\ast }{\omega }_Y^{\otimes m}(aE)$ 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 $f_{\ast }{\omega }_Y(aE)$ to coincide with ${\omega }_X$. We also introduce an index measuring the singularities of normal varieties.