Scattering diagrams, stability conditions, and coherent sheaves on ${\mathbb{P}}^2$

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 ${\mathbb{P}}^2$. This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on ${\mathbb{P}}^2$, or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local ${\mathbb{P}}^2$. As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on ${\mathbb{P}}^2$ is Hodge-Tate, and we give the first non-trivial numerical checks of the general $\chi$-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local ${\mathbb{P}}^2$.

83 pages. Final version published in Journal of Algebraic Geometry

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Cited by in corpus (4)

• Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles
• BPS Dendroscopy on Local $P^2$
• Quivers and curves in higher dimension
• Attractor invariants, brane tilings and crystals