Pure sheaves and Kleinian singularities

TL;DR

This paper extends Grothendieck's decomposition theorem to pure sheaves on Kleinian singularities, exploring classifications and connections to Calabi-Yau categories.

Contribution

It generalizes the decomposition results for pure sheaves beyond type A Kleinian singularities and classifies rigid pure sheaves, linking to spherical objects in Calabi-Yau categories.

Findings

Extended decomposition theorem to other Kleinian singularities.

Classified rigid pure sheaves on fundamental cycles.

Connected pure sheaf classification to Calabi-Yau spherical objects.

Abstract

Grothendieck proved that any locally free sheaf on a projective line over a field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara construct an analogue of Grothendieck's theorem for pure sheaves on the fundamental cycle of the Kleinian singularity $A_n$. We first study the analogue for the other Kleinian singularities except for $A_n$. We also study the classification of rigid pure sheaves on the reduced scheme of the fundamental cycles. The classification is related to the classification of spherical objects in a certain Calabi-Yau $2$-dimensional category.