Scattering diagrams, stability conditions, and coherent sheaves on ℙ²

Bousseau, Pierrick

doi:10.1090/jag/795

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

Author: Pierrick Bousseau
Journal: J. Algebraic Geom. 31 (2022), 593-686
DOI: https://doi.org/10.1090/jag/795
Published electronically: June 24, 2022
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Abstract | References | Additional Information

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 $\mathbb {P}^2$.

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Additional Information

Pierrick Bousseau
Affiliation: Institute for Theoretical Studies, ETH Zurich, 8092 Zurich, Switzerland
MR Author ID: 1306428
Email: pboussea@ethz.ch

Received by editor(s): May 13, 2020
Received by editor(s) in revised form: August 31, 2021
Published electronically: June 24, 2022
Additional Notes: The author acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation.
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