Torsion and cotorsion in the sheaf of K\"ahler differentials on some mild singularities

TL;DR

This paper provides a criterion for when the sheaf of Kähler differentials on certain singularities is torsion-free, revealing that even mild singularities often have torsion and cotorsion in this sheaf.

Contribution

It introduces a new criterion for torsion-freeness of Kähler differentials on cones over smooth varieties and demonstrates torsion presence in mild singularities.

Findings

Sheaf of Kähler differentials can have torsion on mild singularities.

Even Gorenstein terminal singularities exhibit torsion and cotorsion.

The criterion applies to cones over smooth projective varieties.

Abstract

We give a criterion for the sheaf of K\"ahler differentials on a cone over a smooth projective variety to be torsion-free. Applying this to Veronese embeddings of projective space and using known results on differentials on quotient singularities we show that even for mild, e.g. Gorenstein terminal, singularities the sheaf of K\"ahler differentials will in general have torsion and cotorsion.