On direct images of twisted pluricanonical sheaves on normal varieties

TL;DR

This paper investigates the depth properties of direct image sheaves of twisted pluricanonical sheaves on normal varieties, providing conditions under which these sheaves satisfy certain depth criteria and characterizing when they coincide with canonical sheaves.

Contribution

It establishes new depth and isomorphism criteria for direct images of twisted pluricanonical sheaves on normal varieties, introducing an index to measure singularities.

Findings

Higher direct images are $S_2$ under certain conditions.

Criteria for direct images to equal canonical sheaves.

Introduction of a singularity index for normal varieties.

Abstract

We study the depth properties of certain direct image sheaves on normal varieties. Let $f: Y\rightarrow X$ 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 $R^df_*\omega^{\otimes m}_Y(aE)$ 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 $\omega_X$. We also introduce an index measuring the singularities of normal varieties.