2025/06/27 by Pierre Descombes, Descombes, Pierre · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2506.22400
openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove a toric localization formula in the cohomological Donaldson-Thomas theory. Consider a (-1)-shifted symplectic algebraic space with a \mathbbGm-action leaving the (-1)-shifted symplectic form invariant (typical examples are the moduli space of stable sheaves or complexes of sheaves on a Calabi-Yau threefold with a \mathbbGm-invariant Calabi-Yau form or the intersection of two \mathbbGm-invariant Lagrangians in a symplectic space with a \mathbbGm-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 \mathbbGm-fixed components. This result can be seen as a (-1)-shifted version of the Bialynicki-Birula decomposition for smooth schemes. We obtain our result from a similar formula for stacks and Halpern-Leistner's Theta-correspondence, at the level of perverse Nori motives, which we use also to derive foundational constructions in DT theory, in particular the Kontsevich-Soibelman wall crossing formula and the construction of the Cohomological Hall Algebra for smooth projective Calabi-Yau threefolds (a similar construction of the CoHA was also done independently by Kinjo, Park, and Safronov in a recent work). This paper subsumes the previous paper "Hyperbolic localization of the Donaldson-Thomas sheaf" from the same author.