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Mathematics > Algebraic Geometry

arXiv:2201.12215 (math)
[Submitted on 28 Jan 2022 (v1), last revised 11 Oct 2023 (this version, v2)]

Title:Hyperbolic localization of the Donaldson-Thomas sheaf

Authors:Pierre Descombes
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Abstract:In this paper we prove a toric localization formula in cohomological Donaldson Thomas theory. Consider a -1-shifted symplectic algebraic space with a C* action leaving the -1-shifted symplectic form invariant. This includes the moduli space of stable sheaves or complexes of sheaves on a Calabi-Yau threefold with a C*-invariant Calabi-Yau form, or the intersection of two C*-invariant Lagrangians in a symplectic space with a C*-invariant symplectic form. In this case we express the restriction of the Donaldson-Thomas perverse sheaf (or monodromic mixed Hodge module) defined by Joyce et al. to the attracting variety as a sum of cohomological shifts of the DT perverse sheaves on the C* fixed components. This result can be seen as a -1-shifted version of the Bialynicki-Birula decomposition for smooth schemes.
Comments: 45 pages. In this new version, we have simplified the formalism of the proof, insisted more on orientations datas, and removed for clarity the section on stacks
Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th)
Cite as: arXiv:2201.12215 [math.AG]
  (or arXiv:2201.12215v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.2201.12215

Submission history

From: Pierre Descombes [view email]
[v1] Fri, 28 Jan 2022 16:17:04 UTC (60 KB)
[v2] Wed, 11 Oct 2023 16:58:01 UTC (57 KB)
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