Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866915517466411008 |
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| author | Bousseau, Pierrick |
| author_facet | Bousseau, Pierrick |
| contents | 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 $χ$-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local $\mathbb{P}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_02985 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$ Bousseau, Pierrick Algebraic Geometry High Energy Physics - Theory 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 $χ$-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local $\mathbb{P}^2$. |
| title | Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$ |
| topic | Algebraic Geometry High Energy Physics - Theory |
| url | https://arxiv.org/abs/1909.02985 |