Pure sheaves and Kleinian singularities

2017/07/10 by Kotaro Kawatani, Kawatani, Kotaro · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1707.02714

openalex publication_date 2017/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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• Spherical and Semibrick Classifications

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