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arXiv · 1707.02714

Pure sheaves and Kleinian singularities

Kotaro Kawatani
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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.

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Kotaro Kawatani. 2017-07-10. Pure sheaves and Kleinian singularities. https://arxiv.org/abs/1707.02714

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