Hyperbolic localization of the Donaldson-Thomas sheaf

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.