On a Dichotomy for Skyscraper Sheaves under the Bridgeland-King-Reid Equivalence

2025/09/19 by Boris Tsvelikhovskiy, Tsvelikhovskiy, Boris
Engineering · #Seismic Performance and Analysis #Geotechnical Engineering and Underground Structures #Geotechnical Engineering and Analysis

paper · pdf · doi:10.48550/arxiv.2509.15825

Abstract

Let G ⊂ SL₃(ℂ) be a finite abelian subgroup, and let Y = G-Hilb(ℂ³) be the corresponding G-Hilbert scheme. Denote by Ψ: D^b_G(Coh(ℂ³)) → D^b(Coh(Y)) the Bridgeland--King--Reid derived equivalence. For a nontrivial character χ of G, let χ^! be the corresponding skyscraper sheaf supported at the origin. It is known that Ψ(χ^!) is always a pure sheaf supported either in degree 0 or in degree -1. We prove that the proportion of characters χ for which Ψ(χ^!) is supported in degree 0 is a rational number lying between 0.25 and 1, with both bounds being sharp. Moreover, we exhibit families of resolutions for which these proportions attain certain explicit values within this range.

Citations

• The universe inside Hall algebras of coherent sheaves on toric resolutions
• Derived Reid's recipe for abelian subgroups of SL3(C)
• A derived approach to geometric McKay correspondence in dimension three
• Derived categories of toric varieties III
• How to calculate A-Hilb C³
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Related

• Derived Reid's recipe for abelian subgroups of SL3(C)
• How to calculate A-Hilb C³
• Character Sheaves and Depth-zero representations
• Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on ℂ²
• The Hilbert scheme of points on a threefold, I