Scattering diagrams, stability conditions, and coherent sheaves on ℙ²

2019/09/06 by Pierrick Bousseau, Bousseau, Pierrick · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th)

paper · pdf · doi:10.48550/arxiv.1909.02985

openalex publication_date 2019/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on ℙ². This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on ℙ², or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local ℙ². As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on ℙ² is Hodge-Tate, and we give the first non-trivial numerical checks of the general χ-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local ℙ².

Citations

• Grothendieck rings of Artin n-stacks
• Sur la cohomologie de certains espaces de modules de fibrés vectoriels
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Cohomology of quiver moduli, functional equations, and integrality of Donaldson–Thomas type invariants
• A proof of N.Takahashi's conjecture for (ℙ²,E) and a refined sheaves/Gromov-Witten correspondence

Cited by

• Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces

Related

• Generalized Donaldson-Thomas Invariants of 2-Dimensional sheaves on local P²
• Scattering diagrams, sheaves, and curves
• Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks
• New moduli spaces of one-dimensional sheaves on ℙ³
• The geometry of the moduli space of one-dimensional sheaves