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.
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Notes
Though it states for the projective case, one can extend the statement to general case by compactification.
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Chou, CC., Song, L. On direct images of twisted pluricanonical sheaves on normal varieties. manuscripta math. 168, 89–100 (2022). https://doi.org/10.1007/s00229-021-01300-y
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DOI: https://doi.org/10.1007/s00229-021-01300-y